HMMT-Feb 2010 gen, Problem 11

Below is pictured a regular seven-pointed star. Find the measure of angle $a$ in radians.

Solution: The measure of the interior angle of a point of the star is $\frac{\pi}{7}$ because it is an inscribed angle on the circumcircle which intercepts a seventh of the circle ${ }^{1}$ Consider the triangle shown above in bold. Because the sum of the angles in any triangle is $\pi$, \[ 2 \varphi+3\left(\frac{\pi}{7}\right)=\pi=2 \varphi+a \] Canceling the $2 \varphi$ on the right-hand side and on the left-hand side, we obtain \[ a=\frac{3 \pi}{7} \text {. } \] $\fbox{\frac{3 \pi}{7}}$.

Answer: \frac{3 \pi}{7}