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

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

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