How can I find the perimeter of the shaded part?

Question

That is a square intersecting with 2 circles. I am asked to find the perimeter of the shaded part. I found arc length within the square which is $6\pi$. Another thing I can do is 12 + x + x = Perimeter

Answer 1

The radius of each circle is $12$, and the circular portions of the perimeter are arcs that subtend an angle of $\frac{\pi}{3}$ (see the picture).

Each of the circular portions has length $12\cdot\frac{\pi}{3}=4\pi$

Thus the total perimeter is $12+8\pi$.

Answer 2

There are many different ways to do this. Here is one way.

Let $r = 12$ be the common radius of the two circles.

The base of the shaded part is of length $r$.

Now let us find the common length of the two 'curved' sides of the shaded part.

By symmetry, the orthogonal projection of the top corner of the shaded part on its base is exactly the midpoint of the base. Therefore, the cosinus of the angle corresponding to a curved side is $\frac 1 2$ and the angle is $\frac\pi 3$. Thus, each curved side has length $\frac \pi 3 r$ and the total perimeter is $(1 + \frac {2\pi}{3})r$.

For $r = 12$, the perimeter is $12 + 8 \pi$.