Pythagoras' Theorem and Areas

Pythagoras' Theorem

Let's start with a quick refresher of the famous Pythagoras' Theorem.

triangle abc

Pythagoras' Theorem says that, in a right angled triangle:
the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).

a2 + b2 = c2

That means we can draw squares on each side:

Pythagoras Areas

And this will be true:

A + B = C

You can learn more about the Pythagorean Theorem and review its algebraic proof.

A More Powerful Pythagorean Theorem 

Say we want to draw semicircles on each side of a right triangle:

Pythagoras semicircle
A, B and C are the areas of each
semicircle with diameters a, b and c.

Maybe A + B = C ?

But they aren't squares! Yet let's go ahead anyway to see where it leads us.

OK, the area of a circle with diameter "D" is:

Area of Circle1 4  π D2

So the area of a semicircle is half of that:

Area of Semicircle1 8  π D2

And so the area of each semicircle is:

A1 8  πa2

B1 8  πb2

C1 8  πc2

Now our question:

Does A + B = C ?

Let's substitute the values:

Does 1 8 πa2 + 1 8 πb21 8 πc2  ?

We can factor out 1 8 π and we get:

a2 + b2 = c2

Yes! It is simply Pythagoras' Theorem.

Therefore, we have shown that Pythagoras' Theorem is true for semicircles.

Will it work for any other shape?

Pythagoras' Star

Yes! The Pythagorean Theorem can be taken further into a shape-generalized form as long as the shapes are similar (has a special meaning in Geometry).

Shape-Generalization Form of the Pythagorean Theorem:

Given a right triangle, we can draw similar shapes on each side so that the area of the shape constructed on the hypotenuse is the sum of the areas of similar shapes constructed on the legs of the triangle.

A + B = C


  • A is the area of the shape on the hypotenuse.
  • B and C are the areas of the shapes on the legs.

The Theorem still holds for cool shapes that are not polygons, such as this amazing dragon!

Pythagoras' Dragon