Hypotenuse

The hypotenuse is the side of a right triangle that is opposite the right angle. It is the longest side of the triangle. The term comes from Greek, with hypo- meaning “under” or “across” and teinein meaning “to stretch,” reflecting the side that spans the opposite angle. The concept applies only to right triangles; other triangles do not have a hypotenuse.

Pythagoras’ theorem relates the lengths of the three sides: in a right triangle with legs a and b and hypotenuse c, a^2 + b^2 = c^2, so c = sqrt(a^2 + b^2). This relation also underpins distance measurements in the plane: the distance between two points with horizontal and vertical separations Δx and Δy is sqrt((Δx)^2 + (Δy)^2), the hypotenuse of a right triangle formed by those legs.

In trigonometry, the hypotenuse serves as the reference length for defining the basic functions of a right triangle. For an acute angle θ, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. Knowing the hypotenuse and an angle determines the remaining sides via these functions; conversely, knowing the legs defines the angles.

Two common special cases are often cited. In a 45-45-90 triangle, the hypotenuse equals the leg length multiplied by √2. In a 30-60-90 triangle, the sides are in the ratio 1:√3:2, with the hypotenuse twice the shortest leg.

A notable geometric property is that the hypotenuse is the diameter of the triangle’s circumcircle; the midpoint of the hypotenuse is the circle’s center.