The isoperimetric coin puzzle 🪙

A loop of string with the same perimeter lets a rigid coin slip through when it's a circle, yet jams it when it's a square. Drag the perimeter, the coin and the shape to see why: the circle maximizes the inscribed diameter (P/π). Pure canvas + JavaScript, no libraries.

// classic isoperimetric geometry · my own implementation

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setup

loop perimeter (P) 12.0 cm

coin diameter (D) 2.60 cm

loop shape

geometry

inscribed diameter —

coin diameter —

clearance —

enclosed area —

same P: square vs circle

square —

circle —

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💡 the isoperimetric principle in 3 lines

Among all closed shapes with a fixed perimeter P, the circle is the one that encloses the largest area.
The circle also maximizes the minimum width of the opening: its inscribed diameter, P / π.
That's why a rigid coin that jams in a square loop slides right through once that same string is opened into a circle.