Math Notes

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Plane Geometry Solutions

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What is the area of the shaded region on the given figure?
Solution

$$(2r-a)^2 + r^2 = (a+r)^2$$

$$4r^2 - 4ra +a^2 +r^2 = a^2 +2ra +r^2$$

$$4r^2 = 4ra +2ra = 6ra$$

$$a= \frac{2r}{3}$$

The area of the shade region is $$A_s = \pi a^2$$

$$A_s = \pi (\frac{2r}{3})^2$$

$$A_s = \frac{4r^2}{9} \pi$$ Answer