# Pythagorean Theorem: Proof

How to prove the Pythagorean theorem: formula and its proof.

## Formula

For a right triangle,

a^{2} + b^{2} = c^{2}.

This is the Pythagorean theorem.

Let's see the proof of the Pythagorean theorem.

Here's a right triangle.

The legs are a and b.

And the hypotenuse is c.

The plane angle, the dot angle, and the right angle

are the interior angles of the triangle.

So m∠[plane] + m∠[dot] + 90 = 180.

Draw 4 triangles like this.

Recall that

m∠[plane] + m∠[dot] + 90 = 180.

See each side of the whole figure.

The plane angle, the dot angle, and the brown angle

form a line.

So m∠[plane] + m∠[dot] + m∠[brown] = 180.

So m∠[brown] = 90.

So the brown angles are the right angles.

The whole figure is a square.

Its side is (a + b).

So the area of the whole square is

(a + b)^{2}.

Square: Area

This is equal to ...

See one of the right triangle.

The base is b.

The height is a.

So the area of the triangle is

(1/2)ba = (1/2)ab.

There are 4 triangles.

So the sum of the areas of these 4 triangles is

4⋅(1/2)ab.

Plus ...

The center figure is a square.

Its side is c.

So the area of the center square is

c^{2}.

So

(a + b)^{2} = 4⋅(1/2)ab + c^{2}.

(a + b)^{2} = a^{2} + 2ab + b^{2}

Square of a Sum: (a + b)^{2}

4⋅(1/2)ab = 2ab

Cancel +2ab on both sides.

Then a^{2} + b^{2} = c^{2}.

So a^{2} + b^{2} = c^{2}.

This is the proof of the Pythagorean theorem.