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The Roller and the Triangle

Age 14 to 16 Short
Challenge Level

Clare is painting the inside of a short triangular prism with a roller. The cross-section of the prism is an equilateral triangle with side length 1, and the cross-section of the roller is a circle with circumference 1. This is shown in the diagram.

What is the total length that Clare can paint?

Give an exact answer.

This problem is adapted from the World Mathematics Championships
You can find more short problems, arranged by curriculum topic, in our short problems collection.