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 $\sin\left(45^{\circ}\right) = \frac{5}{h}$, and $\sin\left(45^{\circ}\right)$ is $\frac{\sqrt{2}}{2}$. So $\frac{\sqrt{2}}{2} = \frac{5}{h}$. Cross multiply: $\sqrt{2}\cdot h = 10$, and dividing by √2 gives $h = \frac{10}{\sqrt{2}}$, which is $5\sqrt{2}$ 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.