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## #1 2023-06-29 22:45:47

Hannibal lecter
Member
Registered: 2016-02-11
Posts: 388

### Calculate a degree of a triangle

How to calculate a degree of traingle
Sides is 1 length using basic geometry only and Pythagorean theorem without using sine and cosine like we are in 300 BC or 600 BC
I want any simple example
To understand how they were calculate degree in old times
Please this will help me a lot and a lot

Last edited by Hannibal lecter (2023-06-29 22:53:17)

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## #2 2023-06-29 22:51:27

Hannibal lecter
Member
Registered: 2016-02-11
Posts: 388

### Re: Calculate a degree of a triangle

The following is and example but it's without graphs
And figure or draws So I can't see :
Please Mr bob draw for me the steps I want see and imagine
If we are limited to the knowledge and techniques of geometry used in 600 BC, we can still use basic geometric constructions to find the angles of an equilateral triangle with side length 1.

Here's one possible method:

1. Draw an equilateral triangle ABC with side length 1.
2. Draw the circle centered at A with radius AB = AC = 1.
3. Extend the line segment BC to intersect the circle at a point D.
4. Draw the line segment AD, and label the point of intersection with BC as E.
5. Draw the line segment BE.
6. Draw the perpendicular bisector of line segment AB, and label the point of intersection with line segment BE as F.
7. Draw the line segment AF.
8. Draw the perpendicular bisector of line segment DE, and label the point of intersection with line segment AF as G.
9. Draw the line segment BG.
10. Label the point of intersection of line segments AF and BG as H.

Now, we can use the fact that triangle ABC is equilateral to show that angles AHB and AFB are both equal to 120 degrees. This is because triangle ABH is congruent to triangle ABF (by side-side-side), so angle AHB must be congruent to angle AFB.

Next, we can use the fact that line segment AH is the angle bisector of angle BAC to show that angle HAB is equal to angle HAC. This is a property of angle bisectors in triangles.

Finally, we can use the fact that angle HAB and angle HAC are complementary angles to show that each angle is 30 degrees. This is because angle HAB + angle HAC = angle BAC = 60 degrees (since triangle ABC is equilateral), so each angle must be 30 degrees.

Therefore, all three angles of the equilateral triangle with side length 1 are 30 degrees.

Last edited by Hannibal lecter (2023-06-29 22:54:30)

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## #3 2023-06-30 01:16:55

Bob
Registered: 2010-06-20
Posts: 9,719

### Re: Calculate a degree of a triangle

hi HL,

I think the ancient Babylonians invented the angle measure of one degree so that a complete rotation is 360 degrees.  I've not read a definite reason for this but there are two possible explanations: (1) The night sky advances by roughly one degree per night.  This is because of the number of days in a year.  I can explain this more precisely but there's no need to get sidetracked at the moment. (2) 360 has lots of factors, so is a 'nice' number with which to work.

Euclid's book (The Elements) shows, early on, how to show that the angle sum of a triangle is 180 degrees.  If the triangle is equilateral then it has (by definition) rotational symmetry order 3, so each of the three angles must be 360/3 = 60.

3. Extend the line segment BC to intersect the circle at a point D.

But with centre A and AB = AC the line BC 'crosses' the circle at two points, B and C ; so where is D?  From then on I couldn't make any progress with the diagram.

There's a free download of a geometry program called Geogebra.  This will enable you to create your own diagrams.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #4 2023-06-30 01:56:07

Hannibal lecter
Member
Registered: 2016-02-11
Posts: 388

### Re: Calculate a degree of a triangle

Bob wrote:

hi HL,

I think the ancient Babylonians invented the angle measure of one degree so that a complete rotation is 360 degrees.  I've not read a definite reason for this but there are two possible explanations: (1) The night sky advances by roughly one degree per night.  This is because of the number of days in a year.  I can explain this more precisely but there's no need to get sidetracked at the moment. (2) 360 has lots of factors, so is a 'nice' number with which to work.

Euclid's book (The Elements) shows, early on, how to show that the angle sum of a triangle is 180 degrees.  If the triangle is equilateral then it has (by definition) rotational symmetry order 3, so each of the three angles must be 360/3 = 60.

3. Extend the line segment BC to intersect the circle at a point D.

But with centre A and AB = AC the line BC 'crosses' the circle at two points, B and C ; so where is D?  From then on I couldn't make any progress with the diagram.

There's a free download of a geometry program called Geogebra.  This will enable you to create your own diagrams.

Bob

it's a robot answer I don't think it's accurate but it would be useful if it make that examples using graphs and drawing
can you make  a diagram and images using any examples and any lengths of triangle just want to understand how they measure a degree using basic geometry and Pythagoras without using sine and cosine only ratios of side without calling them sine
like Thales and Archimedes did

Last edited by Hannibal lecter (2023-06-30 01:57:38)

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## #5 2023-06-30 06:35:18

Bob
Registered: 2010-06-20
Posts: 9,719

### Re: Calculate a degree of a triangle

I'm confused about what you want me to show.  Are you able to give the complete reference for where you found this?

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #6 2023-07-03 09:13:14

Hannibal lecter
Member
Registered: 2016-02-11
Posts: 388

### Re: Calculate a degree of a triangle

Bob wrote:

I'm confused about what you want me to show.  Are you able to give the complete reference for where you found this?

Bob

Mr bob I want an example to calculate a degree of an triangle with only using basic geometry and trigonometry without using moderns ways like we living in 600BC history, by steps and graphs diagrams please any triangle even with sides length 1

Wisdom is a tree which grows in the heart and fruits on the tongue

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## #7 2023-07-03 21:33:41

Bob
Registered: 2010-06-20
Posts: 9,719

### Re: Calculate a degree of a triangle

Not all triangles can be done like this.

Equilateral is easy, all angles are 60. This follows straight away from the fact that it has rotational symmetry order 3.

If you start with an equilateral, all sides = 2 and split it in half down the middle then each half is 60 30 90.  The sides are 2, 1 and for the third use pythagoras.

There is  a method for an isosceles triangle angles 72, 72 36 by which you can calculate the sides. So, if you knew the sides had that property you could work backwards to get the angles.

Mathematicians have worked out two formulas connecting the sides and the angles of any triangle (SINE RULE and COSINE RULE) but, as the names suggest, these involve trigonometry.  I don't know any other ways to get what you want.

If you have a specific triangle I'll give it a try but may not be able to avoid trig.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #8 2023-07-08 07:04:50

Dennis6185
Novice
Registered: 2023-07-08
Posts: 2

### Re: Calculate a degree of a triangle

Let's consider a right-angled triangle with one side of length 3 units and another side of length 4 units. We want to calculate one of the angles using basic geometry and the Pythagorean theorem.
Apply the Pythagorean theorem: According to the Pythagorean theorem, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, we have:

Hypotenuse² = 3² + 4²
Hypotenuse² = 9 + 16
Hypotenuse² = 25

Taking the square root of both sides, we get:
Hypotenuse = √25
Hypotenuse = 5 units

Determine the angle: Now, we can calculate one of the angles using basic geometry. Since we know the lengths of two sides, we can use the concept of ratios of sides in similar triangles.

In this case, we can consider another right-angled triangle with sides in the ratio of 3:4:5. This triangle is similar to the original triangle, but with all side lengths multiplied by a common factor.

The smaller triangle (3:4:5) is a well-known Pythagorean triple, so we know the angle opposite the side of length 3 is 37 degrees (approximately). Thus, the angle in the original triangle is also approximately 37 degrees.
So, in this example, the angle in the triangle is approximately 37 degrees.

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## #9 2023-07-08 07:05:56

Dennis6185
Novice
Registered: 2023-07-08
Posts: 2

### Re: Calculate a degree of a triangle

https://calculettemauricette.net/

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## #10 2023-07-09 03:26:03

Bob
Registered: 2010-06-20
Posts: 9,719

### Re: Calculate a degree of a triangle

I've been doing some research into this.  This is what I have discovered so far.

If you know the sides and angles for some triangle you can deduce the results for similar triangles (ie. all three angles the same or all three sides enlarged in the same ratio).

I can find no evidence that you can find the angles in any triangle if you know the sides except by some version of trigonometry.

Ptolemy managed to  compute an accurate table of sines and cosines
https://www2.clarku.edu/faculty/djoyce/ … ghist.html

He used theorems established by earlier Greeks such as Hipparchus.

In a recent discovery some think that the Babylonians had a table of sines much earler.
https://theconversation.com/written-in- … blet-81472

The modern method for calculating sines involves power series.

According to Wikipedia https://en.wikipedia.org/wiki/History_of_trigonometry
the Indian mathematician Madhava developed the series computational method
around 1400.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #11 2023-07-16 07:02:18

alirind12
Novice
Registered: 2023-07-12
Posts: 3

### Re: Calculate a degree of a triangle

To calculate the angles of a triangle using basic geometry and the Pythagorean theorem without using sine and cosine, we can use the Law of Cosines. The Law of Cosines allows us to find an angle of a triangle when we know the lengths of all three sides.

The Law of Cosines states that for any triangle with sides of lengths a, b, and c, and angle C opposite side c:

cos(C) = (a^2 + b^2 - c^2) / (2 * a * b)

Using this formula, let's take a simple example:

Consider a triangle with sides of length 3 units, 4 units, and 5 units.

To find angle C (opposite the side of length 5 units), we can apply the Law of Cosines:

cos(C) = (3^2 + 4^2 - 5^2) / (2 * 3 * 4)
cos(C) = (9 + 16 - 25) / (24)
cos(C) = 0 / 24
cos(C) = 0

Now, we need to find the angle C. To do that, we need to find the inverse cosine of 0 (cos⁻¹(0)).

cos⁻¹(0) = 90 degrees

So, angle C is 90 degrees.

For the remaining angles, we can use the Law of Sines or the fact that the sum of the angles in a triangle is always 180 degrees. Since we already found angle C as 90 degrees, the sum of the other two angles is 180 - 90 = 90 degrees.

Suppose the other two angles are A and B.

A + B = 90 degrees

For a simple example, let's assume angle A is 30 degrees. Then,

A + B = 30 + B = 90 degrees
B = 90 - 30 = 60 degrees

So, in this example, the angles of the triangle are 30 degrees, 60 degrees, and 90 degrees.

By using the Law of Cosines and basic geometric principles, ancient mathematicians could calculate angles in triangles without relying on trigonometric functions like sine and cosine.

I hope this example helps you understand how triangles were calculated in ancient times. If you have any further questions or need more examples, feel free to ask!

Regards.

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