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Vamshi Jandhyala

Mathematics

Cozy Circles in Regular Polygons

A puzzle by Xavier Durawa: in a regular polygon, place a circle at the midpoint of each side, tangent to the side and as large as possible without overlap. What fraction of the polygon's area is covered by the n circles?

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A two-line trigonometric argument on two adjacent sides gives the common radius and a clean closed form: π2⋅sin⁡(2π/n)(sin⁡(π/n)+1)2\tfrac{\pi}{2} \cdot \tfrac{\sin(2\pi/n)}{(\sin(\pi/n)+1)^2}. For a triangle this is π37+43\tfrac{\pi\sqrt 3}{7+4\sqrt 3}; for a square, π3+22\tfrac{\pi}{3+2\sqrt 2}. Full derivation and matplotlib drawing code in the PDF.

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