2006 AMC 10B Problem 6

Question

A region is bounded by semicircular arcs constructed on the sides of a square whose sides measure

\tfrac{2}{\pi},

as shown. What is the perimeter of this region?

\dfrac{4}{\pi}

2

\dfrac{8}{\pi}

4

\dfrac{16}{\pi}

Each side of the square is the diameter of one semicircular arc

A semicircle on diameter

d

has arc length

\tfrac12\pi d

Each side has length

\tfrac{2}{\pi},

the diameter of a semicircular arc, so each arc has length

\tfrac12\pi\cdot\tfrac{2}{\pi}=1.

The boundary consists of four such arcs, so the perimeter is

4\cdot1=4.

Thus, the correct answer is D.

Problem 6 in other years