# In or Out?

Weekly Problem 52 - 2014

Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

## Problem

Image

The shaded region shown in the diagram is bounded by four arcs, each of the same radius as that of the surrounding circle.

What fraction of the surrounding circle is shaded?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

## Student Solutions

Let the radii be $r$.

Enclose the circle in a square with sides $2r$.

Image

The unshaded area of the square consists of $4$ quadrants (quarters of circles) of radius $r$.

The total area of the square is $4r^2$ and the area of each quadrant is $\pi r^2/4$.

So the shaded area is $4r^2-\pi r^2=r^2(4-\pi)$.

Therefore the fraction of the circle that is shaded is

$$\frac{r^2(4-\pi)}{\pi r^2}=\frac{4}{\pi}-1\;.$$