hüpotenuusi

Hüpotenuusi is the longest side of a right triangle, located opposite the right angle. In a right triangle with legs a and b and hypotenuse c, the Pythagorean theorem states a^2 + b^2 = c^2, so the hypotenuse length is c = sqrt(a^2 + b^2). Because of this relationship, the hypotenuse is always longer than either leg.

A key geometric property of the hypotenuse is that it is the diameter of the triangle’s circumcircle—the circle that passes through all three vertices of the triangle. This makes the hypotenuse a central element in classical constructions and proofs.

In trigonometry, the hypotenuse serves as the denominator in the basic definitions for a given acute angle theta in a right triangle: sin(theta) = opposite/hypotenuse and cos(theta) = adjacent/hypotenuse. The ratio tangent is the ratio of opposite to adjacent sides and can be derived from these using the hypotenuse.

In coordinate geometry, the length of the hypotenuse can be computed as a Euclidean distance between the endpoints of the legs: c = sqrt((Δx)^2 + (Δy)^2) in the appropriate context. The term is specific to right triangles; non-right triangles do not have a hypotenuse in the same sense.

Etymology traces hüpotenuusi to Greek hypo- “under” and teinein “to stretch,” via Latin hypotenuusa, ultimately yielding the English “hypotenuse.”