LaTeX - A Crash Course

Jai Ganesh · 2022-12-10 03:11:16

Pentagon

Formula of Pentagon

Formulas of the pentagon help us to find out everything about a pentagon shape. From the general formula of polygons, we get the following formula of the pentagons.

Diagonals of a pentagon: =

n \times (n - 3) \div 2 = 5 \times (5 - 3) \div  2 = 5

gives

Sum of interior angles of a pentagon: =

{180}^o \times (n - 2) = {180}^o \times (5 - 2) = {540}^o

gives

Exterior angle of Pentagon:

{540}^o \div n = {540}^o \div 5 = {108}^o

gives

Interior angle of Pentagon:

{360}^o \div n = {360}^o \div 5 = {72}^o

gives

Area of the Pentagon =

1/2 \times \  Perimeter \times Apothem \ sq \ units

gives

Perimeter \ of \ a \ Pentagon = (side1 \ + side2 \ + side3 \ +\  side4 \+ side5) \units.

gives

Jai Ganesh · 2022-12-12 00:47:32

Hexagon

A hexagon is defined as a closed 2D shape that is made up of six straight lines. It is a two-dimensional shape with six sides, six vertices, and six interior angles. The name is divided into 'hex', which means six, and 'gonia', which means corners. Let us learn about hexagon shape in detail in this article.

What is Hexagon?

Hexagon is a two-dimensional geometrical shape that is made of six sides, having the same or different dimensions of length. Some real-life examples of the hexagon shape are a hexagonal floor tile, pencil cross-section, clock, a honeycomb, etc. It can be either regular (with 6 equal side lengths and equal angles) or irregular (with 6 unequal side lengths and angles).

Regular Hexagon

A regular hexagon is defined as a closed 2D shape made up of six equal sides and six equal angles. Each angle of the regular hexagon measures 120 degrees. And the sum of all the interior angles is 120 × 6 = 720 degrees. When it comes to the exterior angles, we know that the sum of exterior angles of any polygon is always 360°. There are 6 exterior angles in a hexagon. So, each of the exterior angles in a regular hexagon measures 360 ÷ 6 = 60 degrees.

A regular hexagon is different from an irregular hexagon as in an irregular hexagon, there is no definite measurement of angles, and the lengths of sides are different. Some of the properties that are common to both irregular and regular hexagons are given below:

* There are 6 sides, 6 interior angles, and 6 vertices in both.
* The sum of all 6 interior angles is always 720 degrees.
* The sum of all 6 exterior angles is always 360 degrees.

Sides of a Hexagon

There are six sides of a hexagon, as shown in the above figure. All are straight edges and form a closed shape. In a regular hexagon, we have six equal sides, while in an irregular hexagon, at least two of the sides of a hexagon are different in measure. If we take the sum of all six sides, we will get the perimeter of the hexagon.

In a regular hexagon, if we know the value of perimeter, then the length of each side can be calculated as "Perimeter ÷ 6". For example, if the perimeter of a regular hexagon is 72 units, then the length of each of the hexagon sides is 72 ÷ 6 = 12 units.

Angles of Hexagon

There are six interior angles and six exterior angles in a hexagon. The sum of all six angles of hexagon is 720 degrees, while the sum of its exterior angles is 360 degrees. Look at some properties related to hexagon angles listed below:

The \ Measurement \ of \ each \ interior \ angle

in \ a \ regular \ hexagon \ is \ {720}^o \ \div 6 = {120}^o.

gives

The \ Measurement \ of \ each \ exterior \ angle

of \ a \ regular \ hexagon \ is \ {360}^o \div \ 6 = {60}^o.

gives

At least two of the angles are of different measurements in an irregular hexagon.

Diagonals of a Hexagon

A diagonal is a segment of a line, that connects any two non-adjacent vertices of a polygon. The number of diagonals of a polygon is given by n(n-3)/2, where 'n' is the number of sides of a polygon. The number of diagonals in a hexagon is given by,

6(6 - 3)/2 = 6(3)/2, \ which \ is \ 9

gives

.

Out of the 9 diagonals, 3 of them pass through the center of the hexagon.

Jai Ganesh · 2022-12-16 22:50:01

Heptagon

Heptagon is a two-dimensional shape with seven angles, seven vertices, and seven edges. This seven-sided polygon “heptagon” is made up of two words ‘Hepta’ and ‘Gonia’, which means seven angles. Another name given to it is septagon or 7-gon. A heptagon has fourteen diagonals. A polygon is a closed two-dimensional shape made up of straight sides having any number of sides. In simple words, we can say that a heptagon is a polygon with 7 sides.

What is a Heptagon?

A heptagon is a seven-sided polygon that has seven angles, seven vertices, and seven edges. They may have the same or different dimensions of length. It is a closed figure and a heptagon with all equal seven sides is called a regular heptagon. Let’s observe the figure given below that shows a heptagon.

Heptagon Sides

The seven sides of a heptagon are straight edges and can be of the same or different lengths. These sides meet each other but do not intersect or cross each other. The heptagon sides meet at the vertices to form a seven-sided closed figure.

Heptagon Angles

A heptagon has seven interior angles and the sum of all interior angles is equal to 900°. Some angles of a the figure can be obtuse or acute. The sum of the exterior angles of a heptagon is equal to 360° and this holds for both regular and irregular heptagons.

Heptagon Diagonals

A heptagon has fourteen diagonals. For a convex heptagon, the diagonals lie inside the figure whereas for a concave heptagon, at least one diagonal lies outside the figure.

Types of Heptagon Shape

Heptagon shapes can be categorized based on their sides and angles.

I) Based on the side lengths, heptagons can be classified as follows:

Regular Heptagon: A regular heptagon is one that has equal sides and equal angles. The sum of all the interior angles of a polygon is equal to

(n - 2) \times {180}^o

gives

, where n is the number of sides. Since a heptagon has 7 sides, the sum of its interior angles is equal to

(7 - 2) \times {180}^o = 5 \times {180}^o = {900}^o

gives

.

The value of each interior angle of a regular heptagon is

{900}^o/7 = {128.57}^o \ approximately

gives

Irregular Heptagon: An irregular heptagon is one that has sides and angles of different measures. The value of each interior angle of an irregular heptagon will be different. However, the sum of all the interior angles of an irregular heptagon is also 900°.

The following figures show a regular and an irregular heptagon.

Regular and Irregular heptagon

II) Based on angle measures, heptagons can be classified as follows:

Convex Heptagon: A convex heptagon has all the interior angles measure less than 180°. They can either be regular or irregular heptagons. All the vertices of the convex heptagon are pointed outwards.

Concave Heptagon: In a concave heptagon at least one of the interior angles is greater than 180°. They can either be regular or irregular heptagons. At least one vertex points inwards in a concave heptagon.

Properties of Heptagon

Now that we know the basic meaning of a heptagon, let us now look into some important properties of a heptagon as follows:

* A heptagon has 7 sides, 7 edges, and 7 vertices.
* The sum of the interior angles of a heptagon is equal to 900°.
* The value of each interior angle of a regular heptagon is equal to 128.57°
* The sum of exterior angles of a heptagon is equal to 360°
* The number of diagonals that can be drawn in a heptagon is 14.
* The measure of the central angle of a regular heptagon is approximately equal to 51.43 degrees.
* A regular heptagon is also known as a convex heptagon since all its interior angles are less than 180°
* An irregular heptagon has unequal sides and angles of different measures.

Regular Heptagon Formula

There are many formulas related to a regular heptagon. Let us understand how to find the perimeter and area of a regular heptagon using the heptagon formulas.

Perimeter of a Heptagon

We know that a regular heptagon has 7 sides of equal length. Therefore, the perimeter of a regular heptagon is given as 7 × Side length. Therefore, the perimeter of a regular heptagon with side length ‘a’ is given as, Perimeter = 7a

Area of a Heptagon

Area of a heptagon is defined as the total space occupied by the polygon. The area of a regular heptagon with side length ‘a’ is calculated using the formula,

Area = (7{a^2}/4) cot (\pi/7)

gives

. This formula can be simplified and approximately written as

{3.634}a^2

gives

, where 'a' is the side length. We can use this to calculate the area of a regular heptagon.

Heptagon Angles

A heptagon consists of 7 interior angles and 7 exterior angles. Let's read about the interior and exterior angles of a heptagon.

Interior Angles of a Regular Heptagon

The sum of interior angles of a regular polygon is given using the interior angle formula that is (n - 2) × 180º where n is the number of sides of the polygon.
Thus, for a heptagon, n = 7. Sum of interior angles of a regular heptagon =

(7 - 2) \times {180}^o  = {900}^o.

gives

Thus, each interior angle of a regular heptagon =

900/7 = {128.57}^o

gives

Exterior Angles of a Regular Heptagon
According to the sum of exterior angles formula, the sum of all the exterior angles of a regular polygon is equal to 360º. Thus, the sum of all the exterior angles of a regular heptagon is equal to 360º. Thus, each exterior angle of a regular heptagon =

360/7 = {51.43}^o

gives

Jai Ganesh · 2022-12-21 01:01:14

Octagon

Octagon is an eight-sided two-dimensional geometrical figure. An octagon consists of 8 interior angles and 8 exterior angles. The sum of the interior angles of an octagon is 1080°, and the sum of its exterior angles is 360°. There are 20 diagonals in an octagon. Octagons are classified into various types based upon their sides and angles. Let us learn more about the octagon shape in this article.

What is an Octagon?

An octagon can be defined as a polygon with eight sides, eight interior angles, and eight vertices. When all the sides and angles of an octagon are equal in measurement, it is called a regular octagon. Every polygon is either convex or concave. Convex octagons bulge outwards, whereas concave octagons have indentations (a deep recess). Convex octagons are those in which all the angles point outwards. A regular octagon is an example of a convex octagon. The octagon in which at least one of its angles points inwards is a concave octagon. Observe the figure given below to see what an octagon looks like.

Octagon Sides

An octagon is a polygon with 8 sides and 8 interior angles. The word 'Octagon' is derived from the Greek word, 'oktágōnon' which means eight angles. That is the reason why it is called an octagon.

Types of Octagons

Depending upon the sides and angles, an octagon is classified into the following categories:

* Regular and Irregular Octagon
* Concave and Convex Octagon

Regular Octagon

The octagon that has eight equal sides and eight equal angles is known as a regular octagon.

In a regular octagon, all the sides are equal in length, and all the angles are equal in measure.

The interior angles add up to 1080° and the exterior angles add up to 360°.

The interior angle at each vertex of a regular octagon is 135°.

Regular Octagon, Irregular Octagon

Irregular Octagon

An octagon in which the sides and angles are not congruent is an irregular octagon. In other words, an irregular Octagon has eight unequal sides and eight unequal angles.

It is an octagon with unequal sides and angles.

All the interior angles are of different measure, but their sum is always 1080º.

Convex Octagon

The octagon in which each interior angle is less than 180° is a convex octagon.

Convex octagons bulge outwards.

None of their interior angles is greater than 180°.

Convex Octagon, Concave Octagon

Concave Octagon

The octagon in which one of the angles points inwards is a concave octagon.

Concave octagons have indentations (a deep recess).

The interior angles are greater than 180°, that is, at least one angle is a reflex angle.

Properties of an Octagon

Here are a few properties of an octagon that can help to identify it easily.

* An octagon is a polygon with eight sides and eight angles.
* All its interior angles sum up to 1080°
* 6 triangles can be formed in a regular octagon with the help of diagonals using a common vertex.
* It has 20 diagonals.

Octagon Diagonals

The diagonal of an octagon is the line segment that connects any two non-adjacent vertices. There are 20 diagonals in an octagon. The formula that is used to find the number of diagonals in any polygon is, Number of diagonals = n(n-3)/2; where 'n' represents the number of sides of the polygon. In this case, there are 8 sides in an octagon. After substituting the value of 'n' = 8 in the formula, we get,

Number \ of \ diagonals = n(n-3)/2 = 8(8 - 3)/2 = (8 \times 5)/2 = 20.

gives

.

Therefore, there are 20 diagonals in an octagon.

Angles of an Octagon

There are 8 interior angles and 8 exterior angles in an octagon. In a regular octagon, each interior angle is 135°. The sum of an octagon's interior angles is 1080°, and the sum of the exterior angles of an octagon is 360°.

The sum of the interior angles of an octagon can be calculated with the help of the following formula where 'n' represents the number of sides (8) in an octagon. Sum of interior angles of a polygon =

(n - 2) \times {180}^o = (8 - 2) \times {180}^o = {1080}^o

gives

.

The sum of the exterior angles of an octagon is 360°.

Angles of a regular Octagon

There are 8 interior angles and 8 exterior angles in an octagon. In a regular octagon, each interior angle is 135°. The sum of an octagon's interior angles is 1080°, and the sum of the exterior angles of an octagon is 360°.

The sum of the interior angles of an octagon can be calculated with the help of the following formula where 'n' represents the number of sides (8) in an octagon. Sum of interior angles of a polygon =

(n - 2) \times {180}^o = (8 - 2) \times {180}^o = {1080}^o.

gives

.

The sum of the exterior angles of an octagon is 360°

In case of an irregular octagon, there is no specific formula to find its area. We divide the octagon into smaller figures like triangles. Then, after calculating the area of all the triangles, we add their areas to get the area of the octagon.

Perimeter of an Octagon

The perimeter of a polygon is the total length of its boundary. In order to calculate the perimeter of an octagon, the length of all the sides should be known. We know that in a regular octagon, all the sides are of equal length. Therefore, the formula that is used to find its perimeter is,

Perimeter of an octagon = Sum of all its sides

Perimeter of a regular octagon = 8a (Where 'a' is the length of one side of the octagon)

Important Notes

An octagon has eight sides.
The sum of all the interior angles in an octagon is always 1080º.
The sum of all the exterior angles in an octagon is always 360º.
A regular octagon has 20 diagonals.
Regular octagons are always convex octagons, while irregular octagons can either be concave or convex.

Jai Ganesh · 2022-12-31 00:58:05

Nonagon

Nonagon is a polygon that has 9 sides, 9 interior angles and 9 exterior angles. A nonagon has 27 diagonals and the sum of the interior angles of a nonagon is 1260°. A nonagon shape can be regular or irregular depending upon the sides and angles. Let us learn more about the nonagon shape.

What is a Nonagon Shape?

A nonagon is a nine-sided geometrical figure which can be regular, irregular, convex or concave depending upon its sides and interior angles.

Nonagon Sides

As seen in the figure given above, a nonagon has 9 sides and the name 'Nonagon' was derived from the Latin words 'nonus' and 'gon' which mean a polygon with nine sides. These 9 sides of a nonagon can be equal or of different measures.

Regular Nonagon

A regular nonagon is one in which all the 9 sides are of equal length and the 9 interior angles are of equal measure. On the other hand, when the sides of a nonagon are of unequal lengths and the angles are of different measure, it is called an irregular nonagon.

Properties of a Regular Nonagon

The following points show the properties of a regular nonagon.

* In a regular nonagon, all the sides are of equal length, and all the interior angles are of equal measure.
* The sum of the interior angles of a regular nonagon is 1260° and the sum of the exterior angles is 360°.
* Each interior angle of a regular nonagon measures 140°.

Convex Nonagon and Concave Nonagon

A convex nonagon has the following properties.

* A convex nonagon is one in which all the interior angles measure less than 180°.
* Since each interior angle of a convex nonagon is less than 180°, convex nonagons seem to bulge outwards.

A concave nonagon has the following properties.

* A concave nonagon is one in which at least one interior angle is a reflex angle.
* In a concave nonagon, the vertices of a concave nonagon can be inwards which gives it that peculiar shape.

Nonagon Angles

There are 9 interior angles and 9 respective exterior angles in a nonagon.

Nonagon Interior Angles

Each interior angle of a regular nonagon measures 140°. This can be calculated using the formula for the interior angles of a regular polygon. Interior angle of a regular polygon =

\dfrac{180(9) - 360}{9} = {140}^o

gives

.

Sum of Interior Angles of a Nonagon

The sum of the interior angles of a nonagon is always 1260°. This sum remains the same irrespective of the nonagon being a regular or an irregular nonagon and this can be calculated using the formula, Sum of interior angles of a polygon =

(n - 2) \times {180}^o

gives

, where 'n' = number of sides of the polygon. Here, after substituting the value of n = 9, we get, Sum of interior angles of a polygon =

(9 - 2) \times {180}^o = {1260}^o

gives

.

Nonagon Exterior Angles

Each exterior angle of a regular nonagon measures 40°. If we observe, we can see that the exterior angle and interior angle form a linear pair of angles, which means they sum up to 180°. Hence, each exterior angle of a regular nonagon will be 180° - 140° = 40°. The sum of the exterior angles of a nonagon is 360°.

Properties of Nonagon

The following properties of a nonagon help to distinguish it from other polygons.

* A nonagon is a polygon with 9 sides, 9 interior angles, and 9 exterior angles.
* The interior angles of a nonagon sum up to 1260°
* A nonagon has 27 diagonals.

Perimeter of Nonagon

The perimeter of a polygon is the total length of its boundary. So, in order to find the perimeter of a nonagon, we need to know the length of all 9 sides. The sum of all the sides gives the perimeter of a nonagon. In case of a regular nonagon, all the sides are of equal length, so we can multiply the length of one side by 9 to get the perimeter. These formulas can be expressed as follows.

Perimeter of a nonagon = Sum of all its sides

Perimeter of a regular nonagon = 9a (Where 'a' is the length of one side of the nonagon).

Jai Ganesh · 2023-01-07 02:16:16

Decagon

In geometry, a decagon is known as a ten-sided polygon or ten-gon. The sum of the interior angles of a simple decagon is 1440° and the sum of the exterior angles of a decagon is 360°.

Meaning of Decagon

A decagon is a ten-sided polygon with ten vertices and ten angles. Thus, a decagon shape can be defined as a polygon having ten sides, ten interior angles, and ten vertices. Based on the sides of a decagon, they are broadly classified into regular decagons and irregular decagons. A regular decagon has 35 diagonals and 8 triangles. The location of these diagonals and triangles is explained in the later sections of this article.

Types of Decagon

Decagons can be categorized as regular and irregular decagons based on side-length and angle measurements. There are three possible classifications of decagon that are given below:

* Regular and Irregular Decagons
* Convex and Concave Decagons
* Simple and Complex Decagons

Regular Decagon

A regular decagon is a polygon along with10 equal sides and 10 vertices. The sides and angles are congruent in a regular decagon. The characteristics of a regular decagon are:

* All sides are equal in length and all the angles are equal in the measure in a regular decagon.
* Each interior angle in regular decagon measures 144º, while each exterior angle measures 36º.

Irregular Decagon

An irregular decagon does not have equal sides and angles. At least two sides and angles are different in measurement.

Convex and Concave Decagons

Like any other polygon, decagons also can be convex and concave. A convex decagon bulges outward as all the interior angles are lesser than 180°. While concave decagons have indentations (a deep recess). At least one of the interior angles is greater than 180° in concave decagons.

Simple and Complex Decagons

Simple decagons refer to decagons with no sides crossing themselves. They follow all of the above said regular decagon rules. While complex decagons refer to decagons that are self-intersecting and have additional interior spaces. They do not strictly follow any prescribed rules of decagons regarding their interior angles and their sums.

Properties of Decagon

Some of the important properties of decagons are listed here.

* The sum of the interior angles is 1440°.
* The sum of the measurements of the exterior angle is 360°.
* The central angle measures 36 degrees in the case of a regular decagon.
* There are 35 diagonals in a decagon.
* There are 8 triangles in a decagon.

Sum of the Interior Angles of Decagon

To find the sum of the interior angles of a decagon, first, divide it into triangles. There are eight triangles in a regular decagon. We know that the sum of the angles in each triangle is 180°. Thus,

{180}^o \times 8 = {1440}^o

gives

Therefore, the sum of all the interior angles of a decagon is 1440°.

We know that the number of sides of a decagon is 10. Hence, we divide the total sum of the interior angles by 10

{1440}^o \div 10 = {144}^o

gives

Thus, one interior angle of a regular decagon shape is 144°. And, the sum of all the interior angles of a decagon is 1440°.

Measure of the Central Angles of a Regular Decagon

To find the measure of the central angle of a regular decagon, we need to draw a circle in the middle. A circle forms 360°. Divide this by ten, because a decagon has 10 sides.

{360}^o \div 10 = {36}^o

gives

.

Thus, the measure of the central angle of a regular decagon is 36°.

Decagon Diagonals

A diagonal is a line that can be drawn from one vertex to another. The number of diagonals of a polygon is calculated by:

n(n-3) \div 2

gives

.

In decagon, n is the number of sides which is equal to 10, so n=10. Now we get,

n(n-3) \div 2 = 10(10-3) \div 2

gives

Thus, the number of diagonals in a decagon is 35.

Jai Ganesh · 2023-01-09 18:25:18

Dodecagon

A dodecagon is a polygon with 12 sides, 12 angles, and 12 vertices. The word dodecagon comes from the Greek word "dōdeka" which means 12 and "gōnon" which means angle. This polygon can be regular, irregular, concave, or convex, depending on its properties.

What is a Dodecagon?

A dodecagon is a 12-sided polygon that encloses space. Dodecagons can be regular in which all interior angles and sides are equal in measure. They can also be irregular, with different angles and sides of different measurements. The following figure shows a regular and an irregular dodecagon.

Types of Dodecagons

Dodecagons can be of different types depending upon the measure of their sides, angles, and many such properties. Let us go through the various types of dodecagons.

Regular Dodecagon

A regular dodecagon has all 12 sides of equal length and all angles of equal measure. It is a 12-sided polygon that is symmetrical.

Irregular Dodecagon

Irregular dodecagons have sides and angles of different measures. There can be an infinite amount of variations. Hence, they all look quite different from each other, but they all have 12 sides. Observe the second dodecagon shown in the figure given above which shows an irregular dodecagon.

Concave Dodecagon

In a concave dodecagon, at least one of its interior angles is greater than 180° and the vertices of a concave dodecagon can be inwards and outwards.

Convex Dodecagon

A dodecagon in which none of its interior angles is greater than 180° and where no vertex points inwards is called a convex dodecagon.

Properties of a Dodecagon

The properties of a dodecagon are listed below which explain about its angles, triangles, and its diagonals.

Interior Angles of a Dodecagon

Each interior angle of a regular dodecagon isagon is equal to 150°. This can be calculated by using the formula:

\dfrac{180n - 360}{n}

gives

where n = the number of sides of the polygon. In a dodecagon, n = 12. Now substituting this value in the formula.

\dfrac{180(12) - 360}{12} = {150}^o

gives

The sum of the interior angles of a dodecagon can be calculated with the help of the formula:

(n - 2) \times {180}^o = (12 - 2) \times {180}^o = {1800}^o

gives

Exterior Angles of a Dodecagon

Each exterior angle of a regular dodecagon is equal to 30°. If we observe the figure given above, we can see that the exterior angle and interior angle form a straight angle (180°). Therefore, 180° - 150° = 30°. Thus, each exterior angle has a measure of 30°. The sum of the exterior angles of a regular dodecagon is 360°.

Diagonals of Dodecagon

The number of distinct diagonals that can be drawn in a dodecagon from all its vertices can be calculated by using the formula:

1/2 \times n \times (n-3)

gives ,

where n = number of sides.

In this case, n = 12. Substituting the values in the formula:

1/2 \times n \times (n-3) = 1/2 \times 12 \times (12-3) = 54

gives

.

Therefore, there are 54 diagonals in a dodecagon.

Jai Ganesh · 2023-01-10 17:04:10

Golden Ratio Part I

The golden ratio, which is often referred to as the golden mean, divine proportion, or golden section, is a special attribute, denoted by the symbol

\phi

gives

, and is approximately equal to 1.618. The study of many special formations can be done using special sequences like the Fibonacci sequence and attributes like the golden ratio.

This ratio is found in various arts, architecture, and designs. Many admirable pieces of architecture like The Great Pyramid of Egypt, Parthenon, have either been partially or completely designed to reflect the golden ratio in their structure. Great artists like Leonardo Da Vinci used the golden ratio in a few of his masterpieces and it was known as the "Divine Proportion" in the 1500s. Let us learn more about the golden ratio in this lesson.

What is the Golden Ratio?

The golden ratio, which is also referred to as the golden mean, divine proportion, or golden section, exists between two quantities if their ratio is equal to the ratio of their sum to the larger quantity between the two. With reference to this definition, if we divide a line into two parts, the parts will be in the golden ratio if:

The ratio of the length of the longer part, say "a" to the length of the shorter part, say "b" is equal to the ratio of their sum "(a + b)" to the longer length.

Refer to the following diagram for a better understanding of the above concept:

\dfrac{a}{b} = \dfrac{a + b}{a} =1.618.... = \phi

gives

Golden ratio definition

It is denoted using the Greek letter ϕ, pronounced as "phi". The approximate value of ϕ is equal to 1.61803398875... It finds application in geometry, art, architecture, and other areas. Thus, the following equation establishes the relationship for the calculation of golden ratio:

\phi = a/b = (a + b)/a = 1.61803398875...

gives

where a and b are the dimensions of two quantities and a is the larger among the two.

Golden Ratio Definition

When a line is divided into two parts, the long part that is divided by the short part is equal to the whole length divided by the long part is defined as the golden ratio. Mentioned below are the golden ratio in architecture and art examples.

There are many applications of the golden ratio in the field of architecture. Many architectural wonders like the Great Mosque of Kairouan have been built to reflect the golden ratio in their structure. Artists like Leonardo Da Vinci, Raphael, Sandro Botticelli, and Georges Seurat used this as an attribute in their artworks.

Golden Ratio Formula

The Golden ratio formula can be used to calculate the value of the golden ratio. The golden ratio equation is derived to find the general formula to calculate golden ratio.

Golden Ratio Equation

From the definition of the golden ratio,

a/b = (a + b)/a = \phi

gives

From this equation, we get two equations:

a/b = ϕ - (1)

gives

(a + b)/a = \phi - (2)

gives

From equation (1),

a/b = \phi

gives

(a + b)/a = \phi - (2)

gives

.

Jai Ganesh · 2023-01-10 17:21:55

Golden Ratio Part II

From equation (1),

a/b = \phi

gives

a = b

Substitute this in equation (2),

(b\phi + b)/b\phi = \phi

gives

b(\phi + 1)/b\phi = \phi

gives

(\phi + 1)/\phi = \phi

gives

1 + 1/\phi = \phi

gives

Jai Ganesh · 2023-01-10 22:32:43

Golden Ratio Part III

Golden Ratio Equation

To calculate the value of the golden ratio is by solving the golden ratio equation.

We know,

\phi = 1 + 1/\phi

gives

Multiplying both sides by

\phi

gives

,

{\phi}^2 = \phi + 1

gives

On rearranging, we get,

{\phi}^2 - \phi -1 = 0

gives

The above equation is a quadratic equation and can be solved using quadratic formula:

\phi = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

gives

Substituting the values of a = 1, b = -1 and c = -1, we get,

\phi = \dfrac{ 1 \pm (1 + 4)}{2}

gives

The solution can be simplified to a positive value giving:

\phi = 1/2 + \sqrt{5}/2

gives

Note that we are not considering the negative value, as

\phi

gives

is the ratio of lengths and it cannot be negative.

Therefore,

\phi = 1/2 + \sqrt{5}/2

gives

.

Jai Ganesh · 2023-01-12 14:27:55

Golden Ratio - Part IV

What is Golden Rectangle?

In geometry, a golden rectangle is defined as a rectangle whose side lengths are in the golden ratio. The golden rectangle exhibits a very special form of self-similarity. All rectangles that are created by adding or removing a square are golden rectangles as well.

Constructing a Golden Rectangle

We can construct a golden rectangle using the following steps:

Step 1: First, we will draw a square of 1 unit. On one of its sides, draw a point midway. Now, we will draw a line from this point to a corner of the other side.

Step 2: Using this line as a radius and the point drawn midway as the center, draw an arc running along the square's side. The length of this arc can be calculated using Pythagoras Theorem:

\sqrt{(1/2)^2 + (1)^2} = \sqrt{5}/2 \ units.

gives

Step 3: Use the intersection of this arc and the square's side to draw a rectangle:

This is a golden rectangle because its dimensions are in the golden ratio. i.e.,

\phi = (\sqrt{{5}/2 + 1/2)/1 \approx 1.61803

gives

.

Jai Ganesh · 2023-01-16 22:05:57

Transcendental Numbers - I

Value of Pi

The value of pi is the ratio of the circumference of a circle to its diameter. The pi symbol is denoted as

\pi

gives

It is also called Archimedes' constant which was named after the Greek mathematician, Archimedes, who created an algorithm to approximate the pi value. The value of pi is irrational, which means that the count of digits after the decimal point is infinite. It is used as either 3.1415929 or 22/7. In geometry, it is used in calculating the surface area, volume, and circumference of various three-dimensional shapes.

What is the Value of Pi?

The value of 'pi' is constant, which means it cannot be changed. It is an irrational number usually approximated to 3.14. It is used in various formulas for the measurement of surface area and volume of various solid shapes. 'Pi' is defined as the ratio of the circumference of the circle to the diameter of the circle. We know that the diameter of a circle is the longest line segment that passes through the center of the circle. Imagine the line of diameter is bent such that it covers a part of the circumference of the circle. Now, π is defined as the total number of times the diameter is wrapped around the circumference of the circle which is 3.14 times approximately.

If we divide the circumference by the diameter, we get approximately 3.14. It should be noted that no matter what size of a circle we draw, the ratio of the circumference to its diameter will always be the same.

Symbol for Pi

The symbol used to denote pi in math is π, which is used in Greek letters. It is known as a mathematical constant as its value does not change. The pi symbol represents 22/7 in fractions or 3.14 (approximately) in decimals.

Formula for Pi

The value of pi can be obtained by dividing the circumference of a circle by its diameter. Since pi is the ratio of the circumference of a circle to its diameter, the formula for calculating

\pi

gives

is:

\pi

gives

=

Circumference/Diameter

gives

One interesting statement that helps us to remember the value of pi is "How I wish I could calculate pi". By counting the number of letters in each word, we can easily write the value of pi. Since 'How' has 3 letters; 'I' - 1 letter, 'wish' - 4 letters, 'I' - 1 letter, 'could' - 5 letters, 'calculate' - 9 letters, 'pi' - 2 letters. Therefore, the value of pi approximated to 6 decimal places is 3.141592.

How to Calculate the Value of Pi?

The value of pi can be calculated with the help of a simple activity. Follow the steps given below to know why the value of pi is the ratio of the circumference to the diameter of the circle:

Step 1: Draw a circle of diameter 1 unit.
Step 2: After this step, take a thread and place it along the border of the circle (the circumference).
Step 3: Now, place the thread on the ruler and note the length.

Repeat the process with diameters of 2 units, 3 units, 4 units, 5 units, and record your observations in the table.

Diameter Circumference Circumference/Diameter
1 unit 3.1 units 3.1/1
2 units 6.2 units (approx.) 6.2/2 = 3.1
3 units 9.3 units (approx.) 9.3/3 = 3.1
4 units 12.4 units (approx.) 12.4/4 = 3.1
5 units 15.5 units (approx.) 15.5/5 = 3.1

We can observe that the ratio of circumference to diameter is always the same which is 3.1.

Value of Pi in Decimals

As discussed above, the value of pi is an irrational number, which means there are infinite decimal places after the number 3. The 100 decimal places of pi consist of all digits from 0 to 9. There are eight 0s, eight 1s, twelve 2s, eleven 3s, ten 4s, eight 5s, nine 6s, eight 7s, twelve 8s, and fourteen 9s. Observe the figure given below which shows that the value of pi in decimals starts with 3 and the 100th decimal digit holds the number 9.

Value of pi in decimals

The value of pi in decimals is non-terminating and non-recurring and is approximated to 100 decimal places as 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679. For ease of calculations, it is often approximated to 3.14.

Value of Pi in Fraction

The value of pi in fraction is 22/7. We know that the value of pi is an irrational number in which the digits after the decimal point are infinite. Therefore, to make calculations easier the value of pi is also defined in the form of a fraction and is expressed as 22/7.

Value of Pi in Degree

The value of pi in degree is 180 degrees, as π radians is equal to 180°. To understand this, we will be looking at the concept of radians which is the SI unit of measuring angles. We know that in a circle, one complete revolution makes the circumference of the circle. It means if we rotate by 360° around a point (known as the center of the circle) at a fixed distance (radius), we will get the circumference, which is equal to

2\pi{r}

gives

, where r denoted the radius of the circle.

So, we can write, Circumference =

2\pi{r} = {360}^{o}

gives

. And, for half a circle, the angle is 180° and the length of the arc will be

2\pi{r}/2 = \pi{r}

gives

. The formula of radians is arc length/radius. So, for half a circle, the arc length is πr and the radius is r.

*

\pi{r}/r \ radians = {180}^o

gives

.

*

\pi \ {radians} = {180}^o

gives

Therefore, the value of pi in degree is 180 degrees.

Important Notes:

Some important points about the pi symbol and value are given below:

* 'Pi' is a mathematical constant that is the ratio of the circumference of a circle to its diameter. It is an irrational number often approximated to 3.14159.
* Based on the problem, for ease of calculation, we use the value of pi as 22/7 or 3.14.

A few basic circle formulas using pi are,
a)

Circumference = \pi \times \ Diameter

gives

b)

{Area} = \pi{r}^2

gives

.

Jai Ganesh · 2023-01-17 16:23:41

Transcendental Numbers - II

In mathematics, we deal with several terminologies with fixed numeric values. These are called constants, and they help in solving mathematical problems with ease.

For example,

\pi \approx 3.14

gives

Likewise, there is a constant e, known as Euler’s number. It was discovered in the 18th century. It is quite interesting to learn about the history of Euler’s number. It is the base of the natural logarithm. The Euler's number digit are e is approximately 2.712818284

The number was first coined by Swiss mathematician Leonhard Euler in the 1720s. John Napier, the inventor of logarithms, made significant contributions to developing the number, while Sebastian Wedeniwski calculated it to 869,894,101 decimal places.

e is called the Euler Number, the Eulerian number, or Napier's Constant. Euler's number proof was first given as that e is an irrational number, so its decimal expansion never terminates.

This mathematical constant is not only used in math but equally important and applicable in physics.

In math, the term e is called Euler's number after the Swiss mathematician Leonhard Euler.

It is mathematically known as Euler's (pronounced as oiler) constant or Napier's constant.

The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of

(1 + 1/n)^n

gives

as n approaches infinity, an expression that arises in the study of compound interest.

{\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .}

gives

It is also the unique positive number a such that the graph of the function y = ax has a slope of 1 at x = 0.

The (natural) exponential function f(x) = ex is the unique function f that equals its own derivative and satisfies the equation f(0) = 1; hence one can also define e as f(1). The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one. There are various other characterizations.

The number e is of great importance in mathematics, alongside

0, 1, \pi, \ and \ i

gives

. All five appear in one formulation of Euler's identity

{\displaystyle e^{i\pi }+1=0}

gives

and play important and recurring roles across mathematics. Like the constant

\pi

gives

, e is irrational (it cannot be represented as a ratio of integers) and transcendental (it is not a root of any non-zero polynomial with rational coefficients). To 50 decimal places, the value of e is:

2.7182818284590452353602874713526624977572470

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms to the base e. It is assumed that the table was written by William Oughtred.

The discovery of the constant itself is credited to Jacob Bernoulli in 1683, the following expression (which is equal to e):

{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.}

gives

The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, writing in a letter to Christian Goldbach on 25 November 1731. Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, while the first appearance of e in publication was in Euler's Mechanica (1736).

Jai Ganesh · 2023-01-18 21:05:26

Transcendental Numbers - III

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,

{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}=\varphi }

gives

where the Greek letter phi

\varphi

gives

or

\phi

gives

denotes the golden ratio. The constant

\varphi

gives

satisfies the quadratic equation

{\displaystyle \varphi ^{2}=\varphi +1}

gives

and is an irrational number with a value of

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=}1.618033988749....

gives

.

The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names.

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of

\varphi

gives

-may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.

Jai Ganesh · 2023-01-19 13:41:04

Icosahedron

An icosahedron is a three-dimensional shape with twenty faces which makes it a polyhedron. It is one of the few platonic solids. In this lesson, we will discuss more about the shape of Icosahedron and formulas related to it with the help of solved examples.

What is Icosahedron?

The word "Icosahedron" is made of two Greek words "Icos" which means twenty and "hedra" which means seat. The icosahedron's definition is derived from the ancient Greek words Icos (eíkosi) meaning 'twenty' and hedra (hédra) meaning 'seat'. It is one of the five platonic solids with equilateral triangular faces. Icosahedron has 20 faces, 30 edges, and 12 vertices. It is a shape with the largest volume among all platonic solids for its surface area. It has the most number of faces among all platonic solids.

Icosahedron's vertices : 12
Icosahedron's faces : 20
Icosahedron's edges : 30

Icosahedron's angles

(a)Angle between edges: 60°
(b)Dihedral angel: 138.19°

Icosahedron's volume formula

(5/12) \times (3+\sqrt{5}) \times a^3

gives

Icosahedron's surface area formula

(5\sqrt{3} × a^2)

gives

.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges congruent), and the same number of faces meet at each vertex. There are only five such polyhedra:

Tetrahedron : Cube : Octahedron : Dodecahedron : Icosahedron
Four faces : Six faces : Eight faces : Twelve faces : Twenty faces.

Jai Ganesh · 2023-01-20 13:06:26

Dodecahedron

Surface area and volume

The surface area A and the volume V of a regular dodecahedron of edge length a are:

{\displaystyle {\displaystyle A=3{\sqrt {25+10{\sqrt {5}}}}a^{2}\approx 20.645\,728\,807a^{2}}}

gives

{\displaystyle V={\frac {1}{4}}(15+7{\sqrt {5}})a^{3}\approx 7.663\,118\,9606a^{3}}

gives

Additionally, the surface area and volume of a regular dodecahedron are related to the golden ratio. A dodecahedron with an edge length of one unit has the properties:

{\displaystyle {\displaystyle A={\frac {15\phi }{\sqrt {3-\phi }}}}}

gives

{\displaystyle {\displaystyle V={\frac {5\phi ^{3}}{6-2\phi }}}}

gives

.

A dodecahedron is a three-dimensional figure having twelve faces that are pentagonal in shape. All the faces are flat 2-D shapes. There are five platonic solids and dodecahedron is one of them. Platonic solids are convex polyhedra in which the faces are made up of congruent regular polygons with the same number of faces meeting at each of their vertices. A dodecahedron is made up of 12 congruent pentagons with 3 pentagonal faces meeting at each of its 20 vertices. There are two types of dodecahedrons - regular and irregular dodecahedrons.

What is a Dodecahedron?

Dodecahedron is derived from the Greek word "dōdeka" means "12" and "hédra" means "face or seat" that shows that it is a polyhedron with 12 sides or 12 faces. Hence, any polyhedra with 12 sides can be called a dodecahedron. It is made up of 12 pentagonal faces.

Regular Dodecahedron

A regular dodecahedron has 12 regular pentagonal sides. You can see in the image of the dodecahedron net shown above that there are 12 pentagonal sides on a dodecahedron.

The surface area of a Dodecahedron

\approx 20.64 \times a^2

gives

square units (where a is the length of one side).

The Volume of a Dodecahedron

\approx 7.66 \times  a^3

gives

cubic units (where a is the length of one side)

Jai Ganesh · 2023-01-21 13:17:08

Octahedron

In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations.

An octahedron is the three-dimensional case of the more general concept of a cross polytope.

Area and volume

The surface area A and the volume V of a regular octahedron of edge length a are:

A=2{\sqrt {3}}a^{2}\approx 3.464a^{2}

gives

V={\frac {1}{3}}{\sqrt {2}}a^{3}\approx 0.471a^{3}

gives

Octahedron

An octahedron is a shape that is formed by joining by two pyramids at its bases. Once joined the shape converts to 8 faced, 12 edges, and 6 vertices. An octahedron is most commonly known as the regular octahedron i.e. when all the faces are of the same shape and size. But in most cases, it is not necessary that all faces have to be the same to be called an octahedron, with the same size also, the shape is still called octahedron. Let us learn more about the meaning, properties, area, and volume formulas of an octahedron.

Meaning of Octahedron

The word octahedron is derived from the Greek word 'Oktaedron' which means 8 faced. An octahedron is a polyhedron with 8 faces, 12 edges, and 6 vertices and at each vertex 4 edges meet. It is one of the five platonic solids with faces that are shaped like an equilateral triangle.

Properties of an Octahedron

Mentioned below are a few properties of an octahedron:

* An octahedron has 6 vertices and at each vertex 4 edges meet.
* An octahedron has 8 faces shaped like an equilateral triangle, in the case of a regular octahedron.
* An octahedron has 12 edges.
* In a regular octahedron, the angles between the edges are measured at 60° but the dihedral angle is measured at 109.28°.
* The formula to calculate the surface area of an octahedron is

2 \times \sqrt{3}\times a^2

gives

.
* The formula to calculate the volume of an octahedron is

\sqrt{2}/3 \times a^3

gives

.

Jai Ganesh · 2023-01-22 14:08:54

Cube - I

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.

The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices.

The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations.

The cube is dual to the octahedron. It has cubical or octahedral symmetry.

The cube is the only convex polyhedron whose faces are all squares.

Formulas

For a cube of edge length a:

Surface area :

6a^{2}

gives

,
Volume :

a^{3}

gives

,
Face diagonal :

\sqrt{2}a

gives

,
Space diagonal :

\sqrt{3}a

gives

Radius of circumscribed sphere :

\frac{\sqrt {3}}{2}}a

gives

Radius of sphere tangent to edges :

\frac{a}{\sqrt {2}}

gives

Radius of inscribed sphere :

{\frac {a}{2}}

gives

Angles between faces (in radians) :

{\frac {\pi }{2}}

gives

.

As the volume of a cube is the third power of its sides

a\times a\times a

gives

, third powers are called cubes, by analogy with squares and second powers.

A cube has the largest volume among cuboids (rectangular boxes) with a given surface area. Also, a cube has the largest volume among cuboids with the same total linear size (length+width+height).

Jai Ganesh · 2023-01-22 21:53:18

Cube - II

A cube is a three-dimensional object that has 6 congruent square faces. Dimensions of all the 6 square faces of the cube are the same. A cube is sometimes also referred to as a regular hexahedron or as a square prism. It is one of the 5 platonic solids. Some real-life examples of a cube are an ice cube, a Rubik's cube, a regular dice, etc.

Cube Definition

A cube is a 3D solid object with six square faces and all the sides of a cube are of the same length. It is also known as a regular hexahedron and is one of the five platonic solids. The shape consists of six square faces, eight vertices, and twelve edges. The length, breadth, and height are of the same measurement in a cube since the 3D figure is a square that has all sides of the same length. In a cube, the faces share a common boundary called the edge that is considered as the bounding line of the edge. The structure is defined with each face being connected to four vertices and four edges, vertex connected with three edges and three faces, and edges are in touch with two faces and two vertices.

Cube Definition

A cube is a 3D solid object with six square faces and all the sides of a cube are of the same length. It is also known as a regular hexahedron and is one of the five platonic solids. The shape consists of six square faces, eight vertices, and twelve edges. The length, breadth, and height are of the same measurement in a cube since the 3D figure is a square that has all sides of the same length. In a cube, the faces share a common boundary called the edge that is considered as the bounding line of the edge. The structure is defined with each face being connected to four vertices and four edges, vertex connected with three edges and three faces, and edges are in touch with two faces and two vertices.

Properties of Cube

A cube is considered a special kind of square prism since all the faces are in the shape of a square and are platonic solid. There are many different properties of a cube just like any other 3D or 2D shape. The properties are:

* A cube has 12 edges, 6 faces, and 8 vertices.
* All the faces of a cube are shaped as a square hence the length, breadth, and height are the same.
* The angles between any two faces or surfaces are 90°.
* The opposite planes or faces in a cube are parallel to each other.
* The opposite edges in a cube are parallel to each other.
* Each of the faces in a cube meets the other four faces.
* Each of the vertices in a cube meets the three faces and three edges.

Cube Formula

The cube formula helps us to find the surface area, diagonals, and volume of a cube. Let us discuss the different formulas of a cube.

Surface Area of a Cube

There are two types of surface areas of a cube - Lateral surface area and Total surface area

Lateral Surface Area of a Cube

The lateral area of a cube is the sum of areas of all side faces of the cube. There are 4 side faces so the sum of areas of all 4 side faces of a cube is its lateral area. The lateral area of a cube is also known as its lateral surface area (LSA), and it is measured in square units.

LSA of a Cube =

4a^2

gives

where a is the side length.

Total Surface Area of a Cube
The total surface area of the cube will be the sum of the area of the base and the area of vertical surfaces of the cube. Since all the faces of the cube are made up of squares of the same dimensions then the total surface area of the cube will be the surface area of one face added five times to itself. It is measured as the "number of square units" (square centimeters, square inches, square feet, etc.). Therefore, the formula to find the surface area of a cube is:

Total Surface Area (TSA) of a Cube =

6a^2

gives

where a is the side length.

Volume of a Cube

The volume of a cube is the space occupied by the cube. The volume of a cube can be found out by finding the cube of the side length of the cube. To determine the volume of a cube, there are different formulas based on different parameters. It can be calculated using the side length or the measure of the cube's diagonal and it is expressed in cubic units of length. Hence, the two different formulas to find the volume of the cube are:

i) The Volume of a Cube (based on side length) =

a^3

gives

where a is the length of the side of a cube
ii) The volume of a Cube (based on diagonal) =

\dfrac{\sqrt{3} \times d^3}{9}

gives

where d is the length of the diagonal of a cube.

Diagonal of a Cube

The diagonal of a cube is a line segment that joins two opposite vertices of a cube. The length of the diagonal of a cube can be determined using the diagonal of a cube formula. It helps in finding the length of the face diagonals and the main diagonals. Each face diagonal forms the hypotenuse of the right-angled triangle formed. A cube has six faces (square-shaped). On each face, there are two diagonals joining the non-adjacent vertices. Therefore, we have twelve face diagonals and four main diagonals that connect the opposite vertices of the cube. The diagonal of a cube formula to calculate the length of a face diagonal and the main body diagonal of a cube is given as,

i) Length of face diagonal of cube =

\sqrt{2}a

gives

units, where a = Length of each side of a cube.
ii) Length of main diagonal of a cube =

\sqrt{3}a

gives

units, where a = Length of each side of a cube.

Jai Ganesh · 2023-01-25 00:18:32

Cuboid

A cuboid is a three-dimensional solid shape that has 6 faces, 8 vertices, and 12 edges. It is one of the most commonly seen shapes around us which has three dimensions: length, width, and height. Sometimes the cuboid shape is confused with a cube since it shares some properties of a cube, however, they are different from each other.

What is a Cuboid?

We know that a rectangle is a two-dimensional shape that has 4 sides. Now, imagine a shape that is formed when many congruent rectangles are placed one on top of the other. The shape thus formed is called a cuboid. Observe the following cuboid which shows its three dimensions: length, width, and height.

Dimensions of a Cuboid

It should be noted that there is no strict rule according to which an edge of a cuboid shape should be named as its length, width (breadth), or height. However, it is understood that if a cuboid is placed flat on a table, then the height represents the length of any vertical edge; the length is taken to be the larger of the two dimensions of the horizontal face of the cuboid, and the width is the smaller of the two dimensions. These dimensions of a cuboid are denoted by 'l' for length, 'w' for width (breadth), and 'h' for height. Apart from these, the face of a cuboid is the flat surface; the edge is the line segment connecting two adjacent vertices, and the vertex is a point at which two or more edges meet.

Cuboid Faces Edges Vertices

Every 3D shape has a definite number of faces, edges, and vertices. A cuboid shape has 6 faces, 12 edges, and 8 vertices. A cuboid has 4 lateral faces and 2 faces of top and bottom. All are in the shape of rectangles. It has 12 edges that include 8 edges of the top and bottom faces and 4 edges that connect them. And, it has 8 vertices which are the vertices of the top and bottom faces. At each vertex, three segments meet from all three dimensions.

Cuboid Formulas

Considering the three main dimensions of a cuboid to be the length (l), width (w), and height (h), observe the basic formulas of a cuboid shape in the following table.

i) Face Diagonals :

\sqrt{l^2 + w^2}

gives

units
ii) Space Diagonals :

\sqrt{(l^2+ w^2 + h^2)}

gives

units
iii) Perimeter :

4(l + w + h)

gives

units
iv) Volume :

(l \times w \times h)

gives

cubic units
v) Surface Area :

2(lw + wh + lh)

gives

square units

Diagonals of a Cuboid

Since a cuboid is a 3D shape, there are two types of diagonals in it:

* Face Diagonals
* Space Diagonals

Diagonals of a Cuboid:

Face Diagonal

Face diagonals can be drawn by connecting the opposite vertices on a particular face of a cuboid and we know that only two diagonals can be drawn on one face of a cuboid. Since a cuboid has 6 faces, a total of 12 face diagonals can be drawn in a cuboid.

Space diagonal

A space diagonal is a line segment that joins the opposite vertices of a cuboid. The space diagonals pass through the interior of the cuboid. Therefore, 4 space diagonals can be drawn inside it.

Surface Area of Cuboid

The total area occupied by a cuboid shape is considered the surface area of a cuboid. Since a cuboid is a 3D figure, the surface area will depend on the length, breadth, and height. It can have two kinds of surface areas - Total surface area and lateral surface area. Hence, the formulae to find the surface area of a cuboid are given below:

i) Total Surface Area of Cuboid,

S = 2 (lb + bh + lh)

gives

square units
ii) Lateral Surface Area of Cuboid,

L = 2h (l + b)

gives

square units

where,

l = Length,
b = Breadth,
h = Height,
S = Total surface area, and
L = Lateral surface area

Volume of Cuboid

The volume of a cuboid is considered the space occupied inside a cuboid. A cuboid's volume depends on its length, breadth, and height. Hence, changing any one of these quantities changes the volume of the shape. The unit of cuboid's volume is given as the cubic units. Therefore, the formula to calculate the volume of a cuboid is:

Volume of a Cuboid =

Base \ Area \times \ Height

gives

The base area for cuboid =

l \times b

gives

Hence, the volume of a cuboid,

V = l \times b \times h = lbh

gives

cubic units.

where,

l = Length
b = Breadth, and
h = Height

Cuboid Properties

The important properties of a cuboid help us to identify a cuboid shape easily. They are as follows:

* A cuboid has 6 faces, 8 vertices, and 12 edges.
* All the angles formed at the vertices of a cuboid are right angles.
* All the faces are rectangular in shape.
* Two diagonals can be drawn on each face of a cuboid.
* The opposite edges are parallel to each other.
* The dimensions of a cuboid are length, width, and height.

Jai Ganesh · 2023-01-27 21:26:36

Tetrahedron

A tetrahedron is a three-dimensional shape that has four triangular faces. One of the triangles is considered as the base and the other three triangles together form the pyramid. The tetrahedron is a type of pyramid, which is a polyhedron with triangular faces connecting the base to a common point and a flat polygon base. It has a triangular base and thus it is also referred to as a triangular pyramid.

Tetrahedron Definition

A tetrahedron is a polyhedron with 4 faces, 6 edges, and 4 vertices, in which all the faces are triangles. It is also known as a triangular pyramid whose base is also a triangle. A regular tetrahedron has equilateral triangles, therefore, all its interior angles measure 60°. The interior angles of a tetrahedron in each plane add up to 180° as they are triangular.

Properties of Tetrahedron

A tetrahedron is a three-dimensional shape that is characterized by some distinct properties.

A tetrahedron has :

* 4 Faces
* 4 Vertices
* 6 Edges

The following are the properties of a tetrahedron which help us identify the shape easily.

* It has 4 faces, 6 edges, and 4 vertices (corners).
* In a regular tetrahedron, all four vertices are equidistant from each other.
* It has 6 planes of symmetry.
* Unlike other platonic solids, it has no parallel faces.
* A regular tetrahedron has four equilateral triangles as its faces.

Surface Area of Tetrahedron

The surface area of a tetrahedron is defined as the total area or region covered by all the faces of the shape. It is expressed in square units, like

m^2, {cm}^2, {in}^2, {ft}^2, {yd}^2,

gives

etc. A tetrahedron can have two types of surface areas:

* Lateral Surface Area of Tetrahedron
* Total Surface Area of Tetrahedron

Lateral Surface Area of a Tetrahedron

The lateral surface area of a tetrahedron is defined as the surface area of the lateral or the slant faces of a tetrahedron. The formula to calculate the lateral surface area of a regular tetrahedron is given as,

LSA of Regular Tetrahedron = Sum of 3 congruent equilateral triangles (i.e. lateral faces)

= 3 \times (\sqrt{3})/4 \  a^2

gives

square units
where a is the side length of a regular tetrahedron.

Total Surface Area of a Tetrahedron

The total surface area of a tetrahedron is defined as the surface area of all the faces of a tetrahedron. The formula to calculate the total surface area of a regular tetrahedron is given as,

TSA of Regular Tetrahedron = Sum of 4 congruent equilateral triangles (i.e. lateral faces)

= 4 \times (\sqrt{3})/4 \ a^2 = \sqrt{3} \ a^2

gives

square units
where a is the side length of the regular tetrahedron.

Volume of Tetrahedron

The volume of a tetrahedron is defined as the total space occupied by it in a three-dimensional plane. The formula to calculate the tetrahedron volume is given as,

The volume of regular tetrahedron

= (1/3) \times area \ of \ the \ base \times height = (1/3) \cdot (\sqrt{3})/4 \cdot a^2 \times (\sqrt{2})/(\sqrt{3}) \times a

gives

= (\sqrt{2})/12 \ a^3

gives

cubic units
where a is the side length of the regular tetrahedron.

Jai Ganesh · 2023-01-28 21:28:32

Triangular Prism

A prism is a solid figure with flat faces, two identical bases, and with the same cross-section all along its length. The name of a particular prism depends on the two bases of the prism which can be triangles, rectangles, or any polygon. For example, a prism with triangular bases is called a triangular prism and a prism with a square base is called a square prism, and so on. A triangular prism has two triangular bases and three rectangular lateral faces. Let us learn more about the triangular prism in this article.

What is a Triangular Prism?

A triangular prism is a 3D shape with two identical faces in the shape of a triangle connected by three rectangular faces. The rectangular faces are referred to as the lateral faces, while the triangular faces are called bases. The bases are also called the top and the bottom (faces) of the prism.

Triangular Prism Meaning: A triangular prism is a 3D polyhedron with three rectangular faces and two triangular faces. The 2 triangular faces are congruent to each other, and the 3 lateral faces which are in the shape of rectangles are also congruent to each other. Thus, a triangular prism has 5 faces, 9 edges, and 6 vertices. Observe the following of a triangular prism in which l represents the length of the prism, h represents the height of the base triangle, and b represents the bottom edge of the base triangle.

Triangular Prism Properties

The properties of a triangular prism help us to identify it easily. Listed below are a few properties of a triangular prism:

* A triangular prism has 5 faces, 9 edges, and 6 vertices.
* It is a polyhedron with 3 rectangular faces and 2 triangular faces.
* The two triangular bases are congruent to each other.
* Any cross-section of a triangular prism is in the shape of a triangle.

Right Triangular Prism

A right triangular prism is a prism in which the triangular faces are perpendicular to the three rectangular faces. In other words, the angle formed at the intersection of triangle and rectangle faces should be 90 degrees, therefore, the triangular faces are perpendicular to the lateral rectangular faces. A right triangular prism has 6 vertices, 9 edges, and 5 faces.

Triangular Prism Faces Edges Vertices

As mentioned above, a triangular prism has 5 faces including 3 lateral rectangular faces and 2 triangular bases, 9 edges, and 6 vertices. The vertices of the triangular prism are the vertices of the two triangular bases connected by lines that form rectangles. Its edges include 6 edges of two triangular bases (3 + 3) and 3 sides that join the bases.

Faces : Edges : Vertices
Triangular Prism : 5 : 9 : 6

Triangular Prism Formulas

There are two important formulae of a triangular prism which are surface area and volume. A brief explanation of both is given below along with the formula.

Surface Area of a Triangular Prism

The surface area of a triangular prism is the area that is occupied by its surface. It is the sum of the areas of all the faces of the prism. Hence, the formula to calculate the surface area is:

Surface \ area = (Perimeter \ of \ the \ base \times Length) + (2 \times Base \ Area) = (a + b + c)L + bh

gives

where,

b is the bottom edge of the base triangle,
h is the height of the base triangle,
L is the length of the prism,
a, b, and c are the three edges (sides) of the base triangle
(bh) is the combined area of the two triangular faces, because

[2 \times (1/2 \times bh)] = bh

gives

.

Volume of a Triangular Prism

The volume of a triangular prism is the product of its triangular base area and the length of the prism. As we already know that the triangular prism base is in the shape of a triangle, the area of the base will be the same as that of a triangle. Hence, the

Volume \ of \ a \ Triangular \ Prism = Area \ of \ base \ triangle \times \ length

gives

or it can also be written as

Volume \ of \ Triangular \ Prism = 1/2 \times b \times h \times l

gives

,
where b is the base length of the triangle, h is the height of the triangle, and l is the length of the prism.

Jai Ganesh · 2023-02-04 17:45:36

Square Pyramid

A square pyramid characterized by a square base is a three-dimensional shape having five faces, thus called a pentahedron. The most famous example of such a square pyramid is the Great Pyramid of Giza. A pyramid is a polyhedron that has a base and 3 or greater triangular faces that meet at a point above the base (the apex). Interestingly, pyramids are named after their base, such as

* Rectangular pyramid
* Triangular pyramid
* Square pyramid
* Pentagonal pyramid
* Hexagonal pyramid

Here, we will explore the concept of a square pyramid and its properties. We will discuss different types of square pyramids along with their formula and the net of the square pyramid for better visualization of its figure. We will solve various examples based on the concept for a better understanding.

What is a Square Pyramid?

A square pyramid is a three-dimensional geometric shape that has a square base and four triangular sides that are joined at a vertex. It is a polyhedron (pentahedron) with five faces. A square pyramid consists of a square base and four triangles connected to a vertex. Its base is a square and the side faces are triangles with a common vertex.

A square pyramid has three components.

The top point of the pyramid is called the apex.
The bottom square is called the base.
The triangular sides are called faces.

Properties of a Square Pyramid

Let us list out the properties we have explored in the above image. All these properties are derived from the definition of a pyramid.

* It has 5 faces.
* It has 4 side faces that are triangles.
* It has a square base.
* It has 5 vertices.
* It has 8 edges.

Types of Square Pyramids

We can distinguish the square pyramids on the basis of the lengths of their edges, position of the apex, and so on. Let us discuss the different types of square pyramids.

Right square pyramid

If the apex of the square pyramid is right above the center of the base, it forms a perpendicular with the base. Such a square pyramid is called the right square pyramid.

Oblique square pyramid

If the apex of the square pyramid is not aligned right above the center of the base, the pyramid is called an oblique square pyramid.

Equilateral square pyramid

If all the triangular faces of a square pyramid have equal edges, then the square pyramid is called an equilateral square pyramid.

Square Pyramid Formula

There are formulas for square pyramids for finding the volume, height, base area, and surface area. Here you can see the formulas of the volume, total surface area (TSA), and lateral surface area (LSA) of the square pyramid.

Base Area of a Square Pyramid

Since the square pyramid has a square base, we can calculate its base area using the same formula as the area of square, which is

{side} \times {side}

or

{base \ edge}^2

gives

or .

Volume of a Square Pyramid

The formula to determine the volume of a square pyramid is:

V = \dfrac{1}{3}a^2{h}

gives

.

Here, a is the length of the base and h is the perpendicular height.

Surface Area of a Square Pyramid

There are two types of surface areas, one is TSA (Total Surface Area), and the other is LSA (Lateral Surface Area). When we talk about its surface area, we generally refer to its total surface area (which is the sum of areas of all faces), whereas the lateral surface area is the sum of the areas of the side faces only. Consider a square pyramid of base edge 'a', height 'h', and slant 'l'.

The formula to calculate the surface area of a square pyramid when its height h and base edge a are given:

Curved Surface Area:

2a\sqrt{\left(\dfrac{a^2}{4} + h^2\right)}

gives

or

2al

gives

.

Total Surface Area

a^2 + 2a\sqrt{\left(\dfrac{a^2}{4} + h^2\right)}

gives

or

a^2 + 2al

gives

.

Important Notes on Square Pyramid

A square pyramid is a three-dimensional geometric shape that has a square base and four triangular sides that are joined at a vertex.

Jai Ganesh · 2023-02-06 22:25:37

Cone

A cone is a three-dimensional shape that has a circular base and it narrows down to a sharp point called a vertex. One of the easiest real-life examples that could be given is a birthday cap in the shape of a cone. With regards to a cone, we have two types of areas. One is the total surface area and the other is a curved surface area. The total surface area of a cone is defined as the area covered by its base and the curved part of the cone, whereas the curved surface area is defined as the area of the curved surfaces of the cone only.

Cone Definition

A cone is a three-dimensional solid geometric shape having a circular base and a pointed edge at the top called the apex. A cone has one face and a vertex. There are no edges for a cone.

The three elements of the cone are its radius, height, and slant height. Radius 'r' is defined as the distance between the center of the circular base to any point on the circumference of the base. Height 'h' of the cone is defined as the distance between the apex of the cone to the center of the circular base. The slant height 'l' is defined as the distance between the apex of the cone to any point on the circumference of the cone. Some of the real-life examples of a cone include a birthday cap, a tent, and a road divider.

Properties of Cone

A cone is a shape that has a curved surface and a circular base. The following properties of a cone help us identify it easily. They are as follows.

* A base of a cone is circular.
* There is one face, one vertex, and no edges for a cone.
* The slant height of a cone is the length of the line segment joining the apex of the cone to any point on the circumference of the base of the cone.
* A cone that has its apex right above the circular base at a perpendicular distance is called a right circular cone.
* A cone that does not have its apex directly above the circular base is an oblique cone.

Cone Formula

There are three important formulas related to a cone. They are the slant height of a cone, the volume of a cone, and its surface area. The slant height of a cone is obtained by finding the sum of the squares of radius and the height of the cylinder which is given by the formula given below.

(slant \ height)^2 = {radius}^2 + {height}^2

gives

If the slant height of the cone is 'l' and the height is 'h' and the radius is 'r', then

l^2 = r^2 + h^2

gives

.

The formula for the slant height of the cone is 'l' =

\sqrt{r^2 + h^2}

gives

.

Curved Surface Area of Cone

The curved surface area of a cone is the area enclosed by the curved part of the cone. For a cone of radius 'r', height 'h', and slant height 'l', the curved surface area is as follows:

Curved \ Surface \ Area = \pi{r}l \ square \ units

gives

.

Total Surface Area of Cone

Total surface area is the sum of the area of the circular base and the area of the curved part of the cone. In other words, it is the sum of the curved surface area of the cone and the area of the circular base, which can be written mathematically as:

Total Surface Area (TSA) = Area of the base (Circle) + Curved Surface Area of the Cone(CSA).

TSA = \pi{r^2} + \pi{r}l \ square \ units

gives

.

Total surface area is sometimes referred to as only surface area. So, whenever we are asked to calculate the surface area of the cone, it means we have to find the total surface area.

Volume of a Cone

Volume = \dfrac{1}{3}\pi{r^2}h \ cubic \ units

gives

.

Let A = Area of base of the cone and h = height of the cone.

Therefore, the volume of cone =

\dfrac{1}{3} \times A \times h

gives

.

Since the base of the cone is circular, we substitute the area to be

\pi{r^2}

gives

.

Volume of cone =

\dfrac{1}{3}\pi{r^2}h \ cubic \ units

gives

.

Also, the volume of a cone is one-third of the volume of a cylinder.

Volume of cone = (1/3) × volume of a cylinder.

Types of Cone

Broadly there are two types of Cones. One is the right circular cone and the other is an oblique cone.

Right Circular Cone : Oblique Cone

A right circular cone has its vertex opposite to the circular base.
An oblique cone does not have its vertex directly opposite to the circular base.

The line representing the height of the cone passes through the center of the base circle and is perpendicular to the radius.
The line representing the height of the cone does not pass through the center of the base circle.

Jai Ganesh · 2023-02-15 01:43:27

Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—parallelepiped and cube in three dimensions, parallelogram and square in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only parallelograms and parallelepipeds exist. Three equivalent definitions of parallelepiped are

* a polyhedron with six faces (hexahedron), each of which is a parallelogram,
* a hexahedron with three pairs of parallel faces, and
* a prism of which the base is a parallelogram.

The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped.

Parallelepipeds are a subclass of the prismatoids.

A parallelepiped is a three-dimensional shape that is formed by six parallelograms. The word 'parallelepiped' is derived from the Greek word parallelepipdon, meaning "a body having parallel bodies". We can say that a parallelepiped relates with a parallelogram just like a cube relates with a square. Parallelepiped has 6 parallelogram-shaped faces, 8 vertices, and 12 edges. Let us understand properties and different formulas associated with a surface area and volume of a parallelepiped in the following sections.

What Is a Parallelepiped?

A parallelepiped is a three-dimensional shape with six faces, that are all in the shape of a parallelogram. It has 6 faces, 8 vertices, and 12 edges. Cube, cuboid, and rhomboid are all special cases of a parallelepiped. A cube is a parallelepiped whose all sides are square-shaped. Similarly, a cuboid and a rhomboid are parallelepipeds with rectangle and rhombus-shaped faces respectively. In the figure given below, we can observe a parallelepiped, with 'a', 'b', and 'c' as side lengths and 'h' as the height of the parallelepiped.

Properties of Parallelepiped

There are certain properties of a parallelepiped that help us distinguish it from other 3-D shapes. These properties are listed below,

* Parallelepiped is a three-dimensional solid shape.
* It has 6 faces, 12 edges, and 8 vertices.
* All faces of a parallelepiped are in the shape of a parallelogram.
* A parallelepiped has 2 diagonals on each face, called the face diagonals. It has a total of 12 face diagonals.
* The diagonals connecting the vertices not lying on the same face are called the body or space diagonal of a parallelepiped.
* Parallelepiped is referred to as a prism with a parallelogram-shaped base.
* Each face of a parallelepiped is a mirror image of the opposite face.

Surface Area of Parallelepiped

The surface area of a parallelepiped is defined as the total area covered by all the surfaces of a parallelepiped. The surface area of a parallelepiped is expressed in square units, like

{in}^2, {cm}^2, m^2, {ft}^2, {yd}^2

gives

, etc. The surface area of parallelepiped can be of two types:

* Lateral Surface Area
* Total Surface Area

Lateral Surface Area of Parallelepiped

The lateral surface area of a parallelepiped is defined as the area of the lateral or side faces of a parallelepiped. To calculate the LSA of a parallelepiped, we need to find the sum of the area covered by the 4 side faces.

Total Surface Area of Parallelepiped

The total surface area of a parallelepiped is defined as the area of all the faces of a parallelepiped. To calculate the TSA of a parallelepiped, we need to find the sum of the area covered by the 6 faces.

Surface Area of Parallelepiped Formula

The formula to calculate the lateral surface area and total surface area of parallelepiped is given as,

LSA \ of \ Parallelepiped = P \times H

gives

TSA \ of \ Parallelepiped = LSA + 2 \times B = (P \times H) + (2 \times B)

gives

where,

B = Base area
H = Height of parallelepiped
P = Perimeter of base

Volume of Parallelepiped

The volume of a parallelepiped is defined as the space occupied by the shape in a three-dimensional plane. The volume of a parallelepiped is expressed in cubic units, like

{in}^3, {cm}^3, {m}^3, {ft}^3, {yd}^3, \ etc

gives

.

Volume of Parallelepiped Formula

Volume of parallelepiped can be calculated using the base area and the height. The formula to calculate the volume of a parallelepiped is given as,

V = B \times H

gives

.
where,

B = Base area
H = Height of parallelepiped.

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