In geometry , a hypotenuse is the longest side of a right triangle , which is always the side opposite the right angle . The length of the hypotenuse of a right triangle can be found using Pythagorean's Theorem , which says that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. For example, if one of the cathets is 3 cm (9 cm² square) and the other is 4 cm (16 cm² square), their squares add up to 25 cm². The length of the hypotenuse is the square root of 25 cm², which is 5 cm.
The word hypotenuse comes from the Greek .eta τὴν ὀρθὴν γωνίαν ὑποτείνουσα hē Ten ORTHEN gōnían hypoteínousa (sc. Γραμμή programs or πλευρά pleura ), it means "side opposite to the right angle" ( APOLLODORUS ). The nominalized participle, ἡ ὑποτείνουσα hē hypoteínousa , was used until the fourth century BC. Used for the hypotenuse of the triangle (documented in Plato , Timaios 54d). The Greek term was borrowed into late Latin in the form hypotēnūsa . The spelling with -e as a hypotenuse is of French origin ( Estienne de La Roche 1520 ).
Calculation of the hypotenuse
A right triangle
The length of the hypotenuse can be calculated using two specified lengths or a length and an acute angle .
If you solve this, you get the formula (under the condition )
with which one can calculate the length of the hypotenuse.
Cathete and height
The height divides a right triangle into two sub-triangles. The base of the height divides the hypotenuse into the hypotenuse sections and . According to the Pythagorean theorem , so . The right triangle is similar to its part triangles because the three interior angles are the same. Therefore, the corresponding aspect ratios match and it applies , so
Many computer languages support the ISO-C standard function hypot(x, y), which returns the above value. The function is designed in such a way that it does not fail even if the simple calculation according to the formula can overflow or underflow, and is also often somewhat more precise.