Cozy Circles in Regular Polygons

A two-line trigonometric argument on two adjacent sides gives the common radius and a clean closed form: $\tfrac{\pi}{2} \cdot \tfrac{\sin(2\pi/n)}{(\sin(\pi/n)+1)^2}$. For a triangle this is $\tfrac{\pi\sqrt 3}{7+4\sqrt 3}$; for a square, $\tfrac{\pi}{3+2\sqrt 2}$. Full derivation and matplotlib drawing code in the PDF.