Question

A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?

$\dfrac{4-\pi}{\pi}$

$\dfrac{1}\pi$

$\dfrac{\sqrt2}{\pi}$

$\dfrac{\pi-1}{\pi}$

$\dfrac{3}\pi$

Video solution:

Click to load, then click again to play

Written solution:

The area of the original circle is $\pi\cdot2^2=4\pi$ .

Join the four quarter-circle endpoints to form a square. The square has diagonals $4$ and $4$ , so its area is $\frac12\cdot4\cdot4=8$ .

The part inside the circle but outside this square has area $4\pi-8$ . Those four pieces are congruent to the pieces inside the square but outside the star.

Thus the star area is $8-(4\pi-8)=16-4\pi$ . The desired ratio is $\dfrac{16-4\pi}{4\pi}=\dfrac{4-\pi}{\pi}$ . Thus, the answer is A .

2012 AMC 8 Problem 24

Video solution:

Written solution:

Problem 24 in Other Years