Angle Constructions

An additional important (compass and straightedge) constructible angle is the $36^\circ$ angle and its relatives, which come up in the construction of the regular pentagon.

Subtracting $36^\circ$ from $60^\circ$ and bisecting $3$ times gets us the $3^\circ$ angle. From this we conclude that every integer multiple of the $3^\circ$ angle is constructible. There are no other constructible angles with an integer number of degrees.

For a characterization of all the constructible angles, we first need to define the Fermat primes. Let $F_n=2^{2^n}+1$. If $F_n$ is prime, it is called a Fermat prime. There are only $5$ known Fermat primes, $3,5,17,257, 65537$.

Theorem: Let $x\ge 0$. The $x^\circ$ degree angle is constructible if and only if $$x=\frac{360q}{2^k P}$$ where $q$ and $k$ are non-negative integers and $P$ is a product (possibly empty) of distinct Fermat primes.