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The tumbler doll

A circle within an equilateral triangle within a circle. What fraction is shaded?

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Solution

Exactly 1/2 of the larger circle is shaded.

From the shown 30-60-90 triangle it is clear that the outer circle has twice the radius of the inner circle. Therefore the small circle covers 1/4 of its area.

The ring outside the small circle, formally called an annulus, covers 3/4. From the threefold rotational symmetry of the figure it is clear that 1/3 of its area is shaded. The total shaded area is 1/4+3/4*1/3=1/2.

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