Hypotenuse.html



A right triangle and its hypotenuse, h, along with catheti, c₁ and c₂.

A hypotenuse is the longest side of a right 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 two other sides.

For example, if one of the other sides has a length of 3 meters (when squared, 9 m²) and the other has a length of 4 m (when squared, 16 m²). Their squares add up to 25 m². The length of the hypotenuse is the square root of this, or 5 m.

The word hypotenuse derives, according to some sources, from the Greek ὑποτείνουσα (hypoteinousa), a combination of hypo- ("under") and teinein ("to stretch") ¹. Others suggest the original meaning in Ancient Greek was for a thing which supports something in the manner of a prop or butress derived from a combination of hypo- ("under") and tenuse ("side"). ²

Calculating the hypotenuse

Usually the length of the hypotenuse is calculated using the square root function in the obvious way. Setting x = c₁ and y = c₂ to avoid subscripts:

In mathematical notation;

Some scientific calculators provide a function to convert from rectangular coordinates to polar coordinates. This gives both the length of the hypotenuse and the angle the hypotenuse makes with the base line (c₁ above) at the same time when given x and y.

See also

• Cathetus
• Triangle geometry
• Space diagonal
• Nonhypotenuse number
• Taxicab geometry
• Trigonometry

Notes

References

• Eric W. Weisstein, Hypotenuse at MathWorld.