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# Funky figures

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The center of a circle is at a vertex of an equilateral triangle as shown. If the area of the circle and the equilateral triangle are both 10, what is the combined area of the two gray regions?

Mar 6, 2023

#1
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The combined area of the two gray regions is 16*sqrt(3) - 4*pi.

Mar 6, 2023
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Mar 6, 2023
#3
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The area of the shaded region is the sum of the circle and triangle minus the white region.

The white region is a 60-degree sector, so its area is $${10 \times {60 \over 360}} = {5 \over 3}$$

This means that the area of the shaded region is $$20 - {5 \over 3 } = {\color{brown}\boxed{55 \over 3}}$$

Mar 6, 2023