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Pythagoras’ Theorem Using Polygons, Circles and Solids

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The theorem of Pythagoras states that for a right-angled triangle with squares constructed on each of its sides, the sum of the areas of the two smaller squares is equal to the area of the largest square.

pythagoras-theorem-using-polygons-circles-and-solids

In the diagram, a, b and c are the side lengths of square A, B and C respectively. Pythagoras’ theorem states that area A + area B = area C, or a2 + b2 = c2.

There are many proofs of the theorem which you might wish to investigate. Our focus will be to see how Pythagoras’ theorem can be applied to shapes other than squares, including three-dimensional solids.

Proof of the Theorem

Pythagoras’ Theorem and Regular Polygons

Pythagoras’ theorem involves areas of squares, which are regular polygons.

A regular polygon is a 2-dimensional (flat) shape where each side has the same length.

Here are the first eight regular polygons.

pythagoras-theorem-using-polygons-circles-and-solids

We can show that Pythagoras’ theorem applies to all regular polygons.

As an example, let’s prove that the theorem is true for regular triangles.

First, construct regular triangles, as shown below.

pythagoras-theorem-using-polygons-circles-and-solids

The area of a triangle with base B and perpendicular height H is (B x H)/2.

To determine the height of each triangle, divide the equilateral triangle into two right-angled triangles and apply Pythagoras’ theorem to one of the triangles.

For triangle A in the diagram, proceed as follows.


pythagoras-theorem-using-polygons-circles-and-solids
pythagoras-theorem-using-polygons-circles-and-solids

We use the same method to find the height of the remaining two triangles.

Hence, the height of triangles A, B and C are respectively

pythagoras-theorem-using-polygons-circles-and-solids

The areas of the triangles are:

pythagoras-theorem-using-polygons-circles-and-solids
pythagoras-theorem-using-polygons-circles-and-solids

We know from Pythagoras’ theorem that a2 + b2 = c2.

Hence, by substitution we have

pythagoras-theorem-using-polygons-circles-and-solids

Or, by expanding the brackets on the left side,

pythagoras-theorem-using-polygons-circles-and-solids

Therefore, area A + area B = area C

Pythagoras’ Theorem With Regular Polygons

To prove the general case that Pythagoras’ theorem is true for all regular polygons, knowledge of the area of a regular polygon is required.

The area of an N-sided regular polygon of side length s is given by

pythagoras-theorem-using-polygons-circles-and-solids

As an example, let’s calculate the area of a regular hexagon.

pythagoras-theorem-using-polygons-circles-and-solids

Using N = 6 and s = 2, we have

pythagoras-theorem-using-polygons-circles-and-solids

Now, to prove that the theorem applies to all regular polygons, align the side of the three polygons with a side of the triangle, such as for the hexagon shown below.

pythagoras-theorem-using-polygons-circles-and-solids

Then we have

pythagoras-theorem-using-polygons-circles-and-solids

Therefore

pythagoras-theorem-using-polygons-circles-and-solids

But again from Pythagoras’ theorem, a2 + b2 = c2.

Hence, by substitution we have

pythagoras-theorem-using-polygons-circles-and-solids

Therefore, area A + area B = area C for all regular polygons.

Pythagoras’ Theorem and Circles

In a similar way, we show that Pythagoras’ theorem applies to circles.

pythagoras-theorem-using-polygons-circles-and-solids

The area of a circle of radius r is πr2, where π is the constant approximately equal to 3.14.

So

pythagoras-theorem-using-polygons-circles-and-solids

But once again, Pythagoras’ theorem states that a2 + b2 = c2.

Hence, by substitution we have

pythagoras-theorem-using-polygons-circles-and-solids

The Three-Dimensional Case

By constructing rectangular prisms (box shapes) using each side of the right-angled triangle, we will show that there is a relationship between the volumes of the three cubes.

pythagoras-theorem-using-polygons-circles-and-solids

In the diagram, k is an arbitrary positive length.

Hence

volume A is a x a x k or a2k

volume B is b x b x k or b2k

volume C is c x c x k or c2k

So volume A + volume B = a2k + b2k = (a2 + b2)k

But from Pythagoras’ theorem, a2 + b2 = c2.

So volume A + volume B = c2k = volume C.

Summary

  • By constructing regular polygons on the sides of a right-angle triangle, Pythagoras’ theorem was used to show that the sum of the areas of the two smaller regular polygons is equal to the area of the largest regular polygon.
  • By constructing circles on the sides of a right-angle triangle, Pythagoras’ theorem was used to show that the sum of the areas of the two smaller circles is equal to the area of the largest circle.
  • By constructing rectangular prisms on the sides of a right-angle triangle, Pythagoras’ theorem was used to show that the sum of the volumes of the two smaller rectangular prisms is equal to the volume of the largest rectangular prism.

A Challenge for You

Prove that when spheres are used, volume A + volume B = volume C.

Hint: The volume of a sphere of radius r is 4πr3/3.

Quiz

For each question, choose the best answer. The answer key is below.

  1. In the formula a^2+b^2=c^2, what does c represent?
    • The shortest side of the right-angled triangle.
    • The longest side of the right-angled triangle.
  2. The two shorter sides of a right-angled triangle are of length 6 and 8. The length of the longest side must be:
    • 10
    • 14
  3. What is the area of a pentagon when each side has length 1 cm?
    • 7 square centimetres
    • 10 square centimetres
  4. The number of sides in a nonagon is
    • 10
    • 9
  5. Choose the correct statement.
    • Pythagoras' theorem can be used for all triangles.
    • If a= 5 and b = 12, then using a^2+b^2=c^2 gives c=13.
    • Not all sides of a regular polygon have to be the same.
  6. What is the area of a circle of radius r?
    • 3.14 x r
    • r/3.14
    • 3.14 x r x r

Answer Key

  1. The longest side of the right-angled triangle.
  2. 10
  3. 7 square centimetres
  4. 9
  5. If a= 5 and b = 12, then using a^2+b^2=c^2 gives c=13.
  6. 3.14 x r x r

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.