Skip to content
Vamshi Jandhyala

Notes

Cozy Circles in Regular Polygons

A nice puzzle by Xavier Durawa. Start with a regular triangle; at the midpoint of each side draw a circle on the inside of the triangle tangent to the side. Each circle has the same radius, maximised so that the circles touch each other. What is the ratio of the three circles’ area to the triangle’s?

Working through an nn-gon in general,

r  =  scos⁡(π/n)2 (sin⁡(π/n)+1),r \;=\; \frac{s \cos(\pi / n)}{2 \, (\sin(\pi / n) + 1)},

and the required ratio is

π2⋅sin⁡(2π/n)(sin⁡(π/n)+1)2.\frac{\pi}{2} \cdot \frac{\sin(2 \pi / n)}{\bigl(\sin(\pi / n) + 1\bigr)^2}.

For a triangle, π3/(7+43)\pi\sqrt 3 / (7 + 4 \sqrt 3). For a square, π/(3+22)\pi / (3 + 2 \sqrt 2).