The idea

Pythagoras' theorem

Why a² + b² = c² holds, spotting the hypotenuse, the add-or-subtract decision, testing whether a triangle is right-angled, distance between two coordinates, and Pythagoras in 3D.

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Geometry & measures · Pythagoras & right-angled trigonometry

Pythagoras' theorem

Why a² + b² = c² holds, spotting the hypotenuse, the add-or-subtract decision, testing whether a triangle is right-angled, distance between two coordinates, and Pythagoras in 3D.

Why it works

The square on the hypotenuse

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides:

a^2 + b^2 = c^2 \qquad (c \text{ is the hypotenuse}).

Here's one way to see it. Take four copies of the right-angled triangle and arrange them inside a square of side

a + b

, leaving a tilted square of side

c

in the middle. The big square's area is

(a+b)^2

; it is also four triangles (

4 \times \frac{1}{2}ab = 2ab

) plus the tilted square (

c^2

). So

(a+b)^2 = 2ab + c^2

, and since

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

, the

2ab

cancels to leave

a^2 + b^2 = c^2

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