Problem 645

The perimeter of the shaded region consists of four pieces: a semi-circle with diameter $PS$, a semi-circle with diameter $PQ$, a semi-circle with diameter $QR$, and a semi-circle with diameter $RS$.

We note that since a circle with diameter $d$ has circumference equal to $\pi d$, then, not including the diameter itself, the length of a semi-circle with diameter $d$ is $\frac{1}{2}\pi d$.

Suppose that the diameter of semi-circle $PQ$ is $x$, the diameter of semi-circle $QR$ is $y$, and the diameter of semi-circle $RS$ is $z$.

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We are given that the diameter of the semi-circle $PS$ is 4.

Since $PQ+QR+RS=PS$, then $x+y+z=4$.

Thus, the perimeter of the shaded region is $$\tfrac{1}{2}\pi (PQ) + \tfrac{1}{2}\pi (QR) + \tfrac{1}{2}\pi (RS) + \tfrac{1}{2}\pi (PS) = \tfrac{1}{2}\pi x + \tfrac{1}{2}\pi y + \tfrac{1}{2}\pi z + \tfrac{1}{2}\pi (4) = \tfrac{1}{2}\pi(x+y+z+4) = \tfrac{1}{2}\pi(4+4) = 4\pi$$ We want to determine the area of the square whose perimeter equals this perimeter (that is, whose perimeter is $4\pi$).

If a square has perimeter $4\pi$, then its side length is $\frac{1}{4}(4\pi) = \pi$, and so its area is $\pi^2$.

Because this problem is multiple choice, then we should get the same answer regardless of the actual lengths of $PQ$, $QR$ and $RS$, since we are not told what these lengths are. Therefore, we can assign to these lengths arbitrary values that satisfy the condition $PQ+QR+RS=4$. For example, if $PQ=QR=1$ and $RS=2$, then we can calculate the perimeter of the shaded region to be $$\tfrac{1}{2}\pi (PQ) + \tfrac{1}{2}\pi (QR) + \tfrac{1}{2}\pi (RS) + \tfrac{1}{2}\pi (PS) = \tfrac{1}{2}\pi (1) + \tfrac{1}{2}\pi (1) + \tfrac{1}{2}\pi (2) + \tfrac{1}{2}\pi (4) = \tfrac{1}{2}\pi(1+1+2+4) = 4\pi$$ and obtain the same answer as above.