Question

In the figure above, the smaller circles each have radius $3$. They are tangent to the larger circle at points $A$ and $C$, and are tangent to each other at point $B$, which is the center of the larger circle. What is the perimeter of the shaded region?

• A. $6\pi$
• B. $8\pi$
• C. $9\pi$
• D. $12\pi$
• E. $15\pi$

Answer 1

Answer: (D)

$12\pi$

To find the perimeter of the shaded region, we need half-circumferences of 3 different circles $—$ the big one (radius $=6$), and two small ones (radius $=3$).

The circumference of a circle is $2\pi r$

So half the circumference of a circle is just $\pi r$.

So the perimeter of the shaded region:

half circumference of big circle $+$ half circumference of small circle $+$ half circumference of small circle

$=$ $\pi \times 6+\pi \times 3+\pi \times 3=6\pi +3\pi +3\pi =12\pi$