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# circles

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In the diagram, each circle is divided into two equal areas and $O$ is the center of the larger circle. The area of the larger circle is $64\pi.$ What is the total perimeter of the shaded regions?

Jul 3, 2022

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Let r be the radius of the larger circle. Because the area is $$64 \pi$$, the radius is 8.

So, the area of the larger shaded region is half the circumference plus the diameter, which is $$\pi r + 2r = 16 + 8 \pi$$.

Now, note that the diameter of the smaller circle is the radius of the larger circle, so the radius of the smaller circle is $$4$$

This means that the perimeter of the shaded region is $$8 + 4 \pi$$, by the formula above.

So, the total perimeter is $$8 + 4 \pi + 16 + 8 \pi = \color{brown}\boxed{24 + 12 \pi}$$

Jul 4, 2022