, 2014

A circle with area $36\pi$ has radius $6$, since the the area of a circle with radius $r$ equals $\pi r^2$ and $\pi (6^2)=36\pi$. The circumference of a circle with radius 6 equals $2\pi(6)=12\pi$. Therefore, each quarter-circle contributes $\frac{1}{4}(12\pi) = 3\pi$ to the circumference. The perimeter of the given figure consists of three quarter-circle sections and two radii from the circle. Thus, its perimeter is $3(3\pi)+2(6)=9\pi+12$.