geometry

Pythagorean Theorem

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².

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The Pythagorean theorem is one of the oldest results in mathematics. For any right triangle with legs of length aa and bb and hypotenuse cc,

a2+b2=c2.a^2 + b^2 = c^2.

It has been proved hundreds of times — by rearrangement, by similarity, by algebra, and by calculus — and generalises to inner-product spaces in higher mathematics.

Direct applications include: computing distances in the plane (the distance formula is just the theorem in disguise), checking whether a triangle is right-angled, navigating in 2D / 3D, and laying out construction.

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