Hypotenuse

In geometry, a hypotenuse (rarely: hypothenuse) is the longest side of a right-angled triangle, the side opposite of the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. For example, if one of the other sides has a length of 3 (when squared, 9) and the other has a length of 4 (when squared, 16), then their squares add up to 25. The length of the hypotenuse is the square root of 25, that is, 5.

The word hypotenuse means essentially "length under", and derives from Latin hypotēnūsa, a transliteration of Ancient Greek hypoteínousa (pleurā́ or grammḗ), the feminine present participle of hypoteínō, a combination of hypó ("under") and teínō ("I stretch" or "length"). The word ὑποτείνουσα was used for the hypotenuse of a triangle by Plato in the Timaeus (dialogue) 54d and by many other ancient authors.

A folk etymology says that tenuse means "side", so hypotenuse means a support like a prop or buttress, but this is inaccurate.

The length of the hypotenuse is calculated using the square root function implied by the Pythagorean theorem. Using the common notation that the length of the two legs of the triangle (the sides perpendicular to each other) are a and b and that of the hypotenuse is c, we have

The Pythagorean theorem, and hence this length, can also be derived from the law of cosines by observing that the angle opposite the hypotenuse is 90° and noting that its cosine is 0:

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