The Swiss cheese

A triangle containing two congruent semicircles. The four tangency points are shown. What fraction is yellow?

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The yellow fraction is 1+(24-14√3)π, which is approximately 21,9%.

Solution by Matthew Arcus.

Clearly a 30° triangle, and the area of a 30° right triangle is A = √3a²/2 where a is length of shortest leg. Putting √3 for radius, we get a = 2+√3, A = √3(2+√3)²/2 and ratio is (A-3π)/A = 1+(24-14√3)π.

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