# How many triangles exist whose angles are rational and side lengths are roots to quadratic equations?

By "rational angles" I mean a rational degree measure (equivalently, angle a rational multiple of $\pi$). Obviously similar triangles should be counted once.

Off the top of my head we have: 30-60-90, equilateral triangles, 45-45-90, and 36-36-72.

• Note that while wendy kriger's answer covers your specific question (values of the form $r+s\sqrt{n}$ with $r$ and $s$ rational and $n$ integer), there are other rational angles whose sines and cosines can be expressed using nestded square roots whose 'leaf values' are rational numbers (or equivalently, which are constructible with ruler and compass). Aug 15 '15 at 4:14