An Exact Hypotenuse

An exact value works below the bar as well as above it. The side opposite a 45° angle is 5, and you want the hypotenuse h.

The ratio is sin(45°) = 5/h, and sin(45°) is √(2)/2. So √(2)/2 = 5/h. Cross multiply: √(2)×h = 10, and dividing by √2 gives h = 10/√(2), which is 5√(2) once the root is moved to the top. As a decimal, about 7.07: longer than 5, as a hypotenuse should be.

For 30° it is easier still: the side opposite 30° is half the hypotenuse, so the hypotenuse is twice that side.