Rectangular Hyperbola is a hyperbola having the transverse axis and the conjugate of 2a units and conjugate axis of 2b units of equal length. The eccentricity of a rectangular hyperbola is

`\sqrt{2}`

gives

, and the equation of a rectangular hyperbola is`x^2 - y^2 = a^2`

gives

**What Is A Rectangular Hyperbola?**

A rectangular hyperbola is a hyperbola having the transverse axis and the conjugate axis of equal length. The arcs of a rectangular hyperbola is the same as the arc of a circle. For a rectangular hyperbola having the transverse axis of length 2a and the conjugate axis of length 2b, we have 2a = 2b, or a = b. The general equation of a rectangular hyperbola is

`x^2 - y^2 = a^2.`

gives

The equation of asymptotes of a rectangular hyperbola is

`y = \pm \ or \ x \ or \ x^2 - y^2 = 0`

gives

The axes or the asymptotes of the rectangular hyperbola are perpendicular to each other. The rectangular hyperbola is related to a hyperbola in a similar form as the circle is related to an ellipse. The eccentricity of a rectangular hyperbola is

`\sqrt{2}`

gives

The graph of the equation y = 1/x is similar to the graph of a rectangular hyperbola.**Properties of Rectangular Hyperbola**

The rectangular hyperbola is similar to a regular hyperbola, and the only difference is the different lengths of the transverse axis and conjugate axis in a hyperbola, and these lengths are equal in a rectangular hyperbola The following are some of the important properties of a rectangular hyperbola.

* The eccentricity of a rectangular hyperbola is equal to

`\sqrt{2}`

gives

* The transverse axis and the conjugate axis in a rectangular hyperbola is of equal length.

* The asymptotoes of a rectangular hyperbola is

`y = \pm \ x \ or \ x^2 - y^2 = 0`

gives

* The asymptotes of a rectangular hyperbola are perpendicular to each other.

* The conjugate of a rectangular hyperbola

`x^2 - y^2 = a^2`

gives

is also a rectangular hyperbola`x^2 - y^2 = -a^2`

gives

* The parametric form of representation of a rectangular hyperbola has the coordinates

`x = aSec\theta, y = aTan\theta`

gives

]]>In mathematics, a hyperbola is an important conic section formed by the intersection of the double cone by a plane surface, but not necessarily at the center. A hyperbola is symmetric along the conjugate axis, and shares many similarities with the ellipse. Concepts like foci, directrix, latus rectum, eccentricity, apply to a hyperbola. A few common examples of hyperbola include the path followed by the tip of the shadow of a sundial, the scattering trajectory of sub-atomic particles, etc.

Here we shall aim at understanding the definition, formula of a hyperbola, derivation of the formula, and standard forms of hyperbola using the solved examples.

**What is Hyperbola?**

A hyperbola, a type of smooth curve lying in a plane, has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. A hyperbola is a set of points whose difference of distances from two foci is a constant value. This difference is taken from the distance from the farther focus and then the distance from the nearer focus. For a point P(x, y) on the hyperbola and for two foci F, F', the locus of the hyperbola is PF - PF' = 2a.

**Hyperbola Definition**

A hyperbola, in analytic geometry, is a conic section that is formed when a plane intersects a double right circular cone at an angle such that both halves of the cone are intersected. This intersection of the plane and cone produces two separate unbounded curves that are mirror images of each other called a hyperbola.

**Parts of a Hyperbola**

Let us check through a few important terms relating to the different parameters of a hyperbola.

Foci of hyperbola: The hyperbola has two foci and their coordinates are F(c, o), and F'(-c, 0).

Center of Hyperbola: The midpoint of the line joining the two foci is called the center of the hyperbola.

Major Axis: The length of the major axis of the hyperbola is 2a units.

Minor Axis: The length of the minor axis of the hyperbola is 2b units.

Vertices: The points where the hyperbola intersects the axis are called the vertices. The vertices of the hyperbola are (a, 0), (-a, 0).

Latus Rectum of Hyperbola: The latus rectum is a line drawn perpendicular to the transverse axis of the hyperbola and is passing through the foci of the hyperbola. The length of the latus rectum of the hyperbola is

`2b^2/a`

gives

.Transverse Axis: The line passing through the two foci and the center of the hyperbola is called the transverse axis of the hyperbola.

Conjugate Axis: The line passing through the center of the hyperbola and perpendicular to the transverse axis is called the conjugate axis of the hyperbola.

Eccentricity of Hyperbola: (e > 1) The eccentricity is the ratio of the distance of the focus from the center of the hyperbola, and the distance of the vertex from the center of the hyperbola. The distance of the focus is 'c' units, and the distance of the vertex is 'a' units, and hence the eccentricity is e = c/a.

**Hyperbola Equation**

The below equation represents the general equation of a hyperbola. Here the x-axis is the transverse axis of the hyperbola, and the y-axis is the conjugate axis of the hyperbola.

`\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1`

gives

.Let us understand the standard form of the hyperbola equation and its derivation in detail in the following sections.

**Standard Equation of Hyperbola**

There are two standard equations of the Hyperbola. These equations are based on the transverse axis and the conjugate axis of each of the hyperbola. The standard equation of the hyperbola is

`\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1`

gives

.has the transverse axis as the x-axis and the conjugate axis is the y-axis. Further, another standard equation of the hyperbola is

`\dfrac{y^2}{a^2} - \drac{x^2}{b^2} = 1`

gives

and it has the transverse axis as the y-axis and its conjugate axis is the x-axis.

**Eccentricity(e) of hyperbola formula:**

`e = \sqrt{1 + \dfrac{b^2}{a^2}}`

gives

.]]>An annulus is an inner region between two concentric circles i.e. two or more circles sharing the same center point. The annulus is shaped like a ring and has many applications in mathematics that we will be learning in this article. Some of the real-life examples are a doughnut, finger rings. etc. Let us learn more about the shape of the annulus and solve a few examples to understand the concept better.

**Annulus Definition**

An annulus is a two-dimensional flat figure shaped in a circular form which is constructed by two concentric circles. The region or the area formed in between these two concentric circles is called the annulus. Since it is a flat figure in a circular form, the edges are two circles with the same center. It is considered a circular disk having a circular hole in the middle.

**Annulus Meaning**

The word annulus is derived from a Latin word, 'annuli', meaning little rings. The shape of the annulus is flat and circular with a hole in between, much like a throw ring or a circular disc. Look at the image below showing two circles i.e. one small circle also called an inner circle and a big circle also called the outer circle. The point which is marked as red is the center of both circles. The shaded colored area, between the boundary of these two circles, is known as an annulus.

**Area of the Annulus**

The annulus area is the area of the ring-shaped space i.e. the enclosed region between the two concentric circles. To calculate the area of the annulus, we need the area of both the inner circle and the outer circle. The dimensions of an annulus are defined by the two radii R, and r, which are the radii of the outer ring and the inner ring respectively. Once the measurements of both the radii are known, we can calculate the area by subtracting the area of the small circle from the big circle. Hence, the formula used for finding the area of the annulus is:

Area of Outer Circle =

`\pi{R}^2`

gives

.Area of Inner Circle =

`\pi{r}^2`

gives

.Area of Annulus = Area of Outer Circle – Area of Inner Circle

Therefore, Area of Annulus =

`\pi(R^2 - r^2)`

gives

square units, or it can be written as

`\pi(R + r)(R - r)`

gives

square units, where R is the radius of the outer circle, r is the radius of the inner circle, and π(pi) is approximately 3.142.

The area of the outer (bigger) circle - the area of the inner (smaller) circle = the area of the annulus.

**Annulus Perimeter**

The perimeter is the distance around the 2D shape. Since the annulus is a flat circular shape constructed by two concentric circles, it can also be considered as a ring. Therefore, an open ring can be considered as the topological equivalent of a cylinder and a punctured plane. Similar to the area, to find the perimeter of the annulus we need to consider both the inner circle and the outer circle. So, the perimeter of the ring or annulus is equal to the sum of the radii of the large and small circles multiplied by 2π. The formula for finding the perimeter is:

Perimeter of Annulus (P) =

`2\pi(R + r)`

gives

units, where R is the radius of the outer circle, r is the radius of the inner circle, and`\pi`

gives

(pi) is approximately 3.142.]]>In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same.

`{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}`

gives

**Ellipse**

Ellipse is an integral part of the conic section and is similar in properties to a circle. Unlike the circle, an ellipse is oval in shape. An ellipse has an eccentricity less than one, and it represents the locus of points, the sum of whose distances from the two foci of the ellipse is a constant value. A simple example of the ellipse in our daily life is the shape of an egg in a two-dimensional form and the running tracking in a sports stadium.

Here we shall aim at knowing the definition of an ellipse, the derivation of the equation of an ellipse, and the different standard forms of equations of the ellipse.

**What is an Ellipse?**

An ellipse in math is the locus of points in a plane in such a way that their distance from a fixed point has a constant ratio of 'e' to its distance from a fixed line (less than 1). The ellipse is a part of the conic section, which is the intersection of a cone with a plane that does not intersect the cone's base. The fixed point is called the focus and is denoted by S, the constant ratio 'e' as the eccentricity, and the fixed line is called as directrix (d) of the ellipse.

**Ellipse Definition**

An ellipse is the locus of points in a plane, the sum of whose distances from two fixed points is a constant value. The two fixed points are called the foci of the ellipse.

The general equation of an ellipse is used to algebraically represent an ellipse in the coordinate plane. The equation of an ellipse can be given as,

`\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1.`

gives

**Parts of an Ellipse**

Let us go through a few important terms relating to different parts of an ellipse.

* Focus: The ellipse has two foci and their coordinates are F(c, o), and F'(-c, 0). The distance between the foci is thus equal to 2c.

* Center: The midpoint of the line joining the two foci is called the center of the ellipse.

* Major Axis: The length of the major axis of the ellipse is 2a units, and the end vertices of this major axis is (a, 0), (-a, 0) respectively.

* Minor Axis: The length of the minor axis of the ellipse is 2b units and the end vertices of the minor axis is (0, b), and (0, -b) respectively.

* Latus Rectum: The latus rectum is a line drawn perpendicular to the transverse axis of the ellipse and is passing through the foci of the ellipse. The length of the latus rectum of the ellipse is

`{2b^2}/a`

gives

.* Transverse Axis: The line passing through the two foci and the center of the ellipse is called the transverse axis.

* Conjugate Axis: The line passing through the center of the ellipse and perpendicular to the transverse axis is called the conjugate axis

* Eccentricity: (e < 1). The ratio of the distance of the focus from the center of the ellipse, and the distance of one end of the ellipse from the center of the ellipse. If the distance of the focus from the center of the ellipse is 'c' and the distance of the end of the ellipse from the center is 'a', then eccentricity e = c/a.

**Standard Equation of an Ellipse**

There are two standard equations of the ellipse. These equations are based on the transverse axis and the conjugate axis of each of the ellipse. The standard equation of the ellipse

`\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1`

gives

has the transverse axis as the x-axis and the conjugate axis as the y-axis. Further, another standard equation of the ellipse is

`\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1`

gives

and it has the transverse axis as the y-axis and its conjugate axis as the x-axis. The below image shows the two standard forms of equations of an ellipse.

**Area of Ellipse Formula**

The area of an ellipse is defined as the total area or region covered by the ellipse in two dimensions and is expressed in square units like

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

gives

etc. The area of an ellipse can be calculated with the help of a general formula, given the lengths of the major and minor axis. The area of ellipse formula can be given as,Area of ellipse =

`\pi{a}{b}`

gives

where,

a = length of semi-major axis

b = length of semi-minor axis

**Eccentricity of an Ellipse Formula**

Eccentricity of an ellipse is given as the ratio of the distance of the focus from the center of the ellipse, and the distance of one end of the ellipse from the center of the ellipse

Eccentricity of an ellipse formula,

`e = \dfrac{c}{a} = \sqrt{1 - \dfrac{b^2}{a^2}}`

gives

**Latus Rectum of Ellipse Formula**

Latus rectum of of an ellipse can be defined as the line drawn perpendicular to the transverse axis of the ellipse and is passing through the foci of the ellipse. The formula to find the length of latus rectum of an ellipse can be given as,

`L = {2b^2}/a`

gives

**Formula for Equation of an Ellipse**

The equation of an ellipse formula helps in representing an ellipse in the algebraic form. The formula to find the equation of an ellipse can be given as,

Equation of the ellipse with centre at (0,0) :

`x^2/a^2 + y^2/b^2 = 1`

gives

Equation of the ellipse with centre at (h,k) :

`(x-h)^2 /a^2 + (y-k)^2/ b^2 =1`

gives

.]]>In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.[a]

The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar.

Parabolas have the property that, if they are made of material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas.

The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. It is frequently used in physics, engineering, and many other areas.

**Parabola**

A parabola is a graph of a quadratic function. Pascal stated that a parabola is a projection of a circle. Galileo explained that projectiles falling under the effect of uniform gravity follow a path called a parabolic path. Many physical motions of bodies follow a curvilinear path which is in the shape of a parabola. In mathematics, any plane curve which is mirror-symmetrical and usually is of approximately U shape is called a parabola. Here we shall aim at understanding the derivation of the standard formula of a parabola, the different standard forms of a parabola, and the properties of a parabola.

**What is Parabola?**

A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed line. The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola. Also, an important point to note is that the fixed point does not lie on the fixed line. A locus of any point which is equidistant from a given point (focus) and a given line (directrix) is called a parabola. Parabola is an important curve of the conic sections of the coordinate geometry.

**Parabola Equation**

The general equation of a parabola is:

`y = a(x-h)^2 + k \ or \ x = a(y-k)^2 +h`

gives

, where (h,k) denotes the vertex.The standard equation of a regular parabola is

`y^2 = 4ax`

gives

.Some of the important terms below are helpful to understand the features and parts of a parabola.

* Focus: The point (a, 0) is the focus of the parabola

* Directrix: The line drawn parallel to the y-axis and passing through the point (-a, 0) is the directrix of the parabola. The directrix is perpendicular to the axis of the parabola.

* Focal Chord: The focal chord of a parabola is the chord passing through the focus of the parabola. The focal chord cuts the parabola at two distinct points.

* Focal Distance: The distance of a point

`(x_1,y_1)`

gives

on the parabola, from the focus, is the focal distance. The focal distance is also equal to the perpendicular distance of this point from the directrix.* Latus Rectum: It is the focal chord that is perpendicular to the axis of the parabola and is passing through the focus of the parabola. The length of the latus rectum is taken as LL' = 4a. The endpoints of the latus rectum are (a, 2a), (a, -2a).

Eccentricity: (e = 1). It is the ratio of the distance of a point from the focus, to the distance of the point from the directrix. The eccentricity of a parabola is equal to 1.

**Standard Equations of a Parabola**

There are four standard equations of a parabola. The four standard forms are based on the axis and the orientation of the parabola. The transverse axis and the conjugate axis of each of these parabolas are different. The below image presents the four standard equations and forms of the parabola.

The following are the observations made from the standard form of equations:

* Parabola is symmetric with respect to its axis. If the equation has the term with

`y^2`

gives

, then the axis of symmetry is along the x-axis and if the equation has the term with`x^2`

gives

, then the axis of symmetry is along the y-axis.* When the axis of symmetry is along the x-axis, the parabola opens to the right if the coefficient of the x is positive and opens to the left if the coefficient of x is negative.

* When the axis of symmetry is along the y-axis, the parabola opens upwards if the coefficient of y is positive and opens downwards if the coefficient of y is negative.

**Parabola Formula**

Parabola Formula helps in representing the general form of the parabolic path in the plane. The following are the formulas that are used to get the parameters of a parabola.

* The direction of the parabola is determined by the value of a.

* Vertex = (h,k), where h = -b/2a and k = f(h)

* Latus Rectum = 4a

* Focus: (h, k+ (1/4a))

* Directrix: y = k - 1/4a

**Derivation of Parabola Equation**

Let us consider a point P with coordinates (x, y) on the parabola. As per the definition of a parabola, the distance of this point from the focus F is equal to the distance of this point P from the Directrix. Here we consider a point B on the directrix, and the perpendicular distance PB is taken for calculations.

As per this definition of the eccentricity of the parabola, we have PF = PB (Since e = PF/PB = 1)

The coordinates of the focus is F(a,0) and we can use the coordinate distance formula to find its distance from P(x, y)

`PF = \sqrt{(x - a)^2 + (y - 0)^2} = \sqrt{(x - a)^2 + y^2}`

gives

The equation of the directrix is x + a = 0 and we use the perpendicular distance formula to find PB.

Squaring the equation on both sides,

`(x - a)^2 + y^2 = (x + a)^2`

gives

`x^2 + a^2 - 2ax + y^2 = x^2 + a^2 + 2ax`

gives

`y^2 - 2ax = 2ax`

gives

`y^2 = 4ax`

gives

]]>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.

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.

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.

]]>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.]]>

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 unitswhere 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 unitswhere 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 unitswhere a is the side length of the regular tetrahedron.]]>

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

unitsii) Space Diagonals :

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

gives

unitsiii) Perimeter :

`4(l + w + h)`

gives

unitsiv) Volume :

`(l \times w \times h)`

gives

cubic unitsv) 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 unitsii) Lateral Surface Area of Cuboid,

`L = 2h (l + b)`

gives

square unitswhere,

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.

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 cubeii) 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.]]>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).

]]>