2006 AMC 10B Problem 6

1. First decide what the perimeter is. The outside edge of the region is made only of the four curved arcs. The square's sides are inside the shaded shape, so they do not count.
2. Each arc is half of a circle drawn on one side of the square. That side goes straight across the circle, so it is the diameter: $d = \frac{2}{\pi}$.
3. A full circle has circumference $\pi \times d$. Here that is
$$\pi \cdot \frac{2}{\pi} = 2.$$
4. Each arc is half a circle, so its length is $2 \div 2 = 1$.
5. There are four arcs, so the perimeter is $4 \times 1 = 4$.

The answer is $\boxed{\textbf{(D)}\ 4}$.

The strange side length $\frac{2}{\pi}$ is a hint: it was picked so the $\pi$ in the circumference formula cancels and the arcs come out to whole numbers. Perimeter means only the outside boundary, so always check which lines are really on the edge of the shape.

The boundary of the region consists only of the four semicircular arcs; the square's sides are interior to the region.

Each arc is built on a side of the square, so that side is the arc's diameter, $d = \frac{2}{\pi}$. A full circle of diameter $d$ has circumference $\pi d$, so a semicircle has arc length
$$\frac{\pi d}{2} = \frac{\pi}{2} \cdot \frac{2}{\pi} = 1.$$

Four arcs give a perimeter of $4 \cdot 1 = 4$.

The answer is $\boxed{\textbf{(D)}\ 4}$.