S2 · Geometry · GEO.PYTH

Apply Pythagorean theorem

In a right triangle, , where c is the hypotenuse.

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Intuition

Understand first, compress later.

The areas of the squares on the two legs add exactly to the area of the square on the hypotenuse.

Switch representations

Pythagoras' theorema = 3, b = 4, c = 5

square on leg asquare on leg bsquare on the hypotenuse cright angle

The drawing is to scale: the legs measure 3 and 4, the hypotenuse exactly 5. The two squares on the legs together cover exactly as much area as the square on the hypotenuse: 9 + 16 = 25. Count the unit squares — they come out exactly even. This holds only because the angle between the legs is a right angle; the small square in the corner says exactly that.

Formal view

First identify the hypotenuse opposite the right angle. Substitute side lengths and solve for the unknown length.

Why does it work?

The theorem describes a fixed metric relationship that applies specifically to right triangles.