Question

In the figure, O, P and R are the centers of three circles, each with radius 2 What is the perimeter of the shaded region?

• A. 49.78
• B. 54
• C. 56.52
• D. 60.34

Answer 1

The correct option is C 56.52

Assume a $△OPR$

in which $OP=PR=OR=r=2$

Hence, we can see that $△OPR$ is Equilateral (with each angle = ${60}^o$)

also, the length of $arc\left(OP\right)=\frac{{60}^o}{{360}^o}\ast (2\pi \ast 2)=\frac{2\pi }{3}=18.84$

Similarly, we can see that $arc\left(OR\right),arc\left(RP\right)$ subtend equal angles($={60}^o$)

So,$arc\left(OP\right)=arc\left(OR\right)=arc\left(PR\right)=18.84$

Hence, Perimeter of shaded Region $=3\ast 18.84=56.52$