#
Problem
of the Week

Problem
C and Solution

A
Divided Circle

## Problem

Two circles are said to be *concentric* if they have the same
centre.

In the diagram below, there are two concentric circles. The radius of
the smaller circle is \(5\) cm. The
portion inside the larger circle that is outside of the smaller circle
is divided into eight congruent pieces that each have the same area as
the smaller circle.

In terms of \(\pi\), what is the
circumference of the larger circle?

## Solution

Since the radius of the smaller circle is \(5\) cm, it follows that the area of this
circle is \(\pi (5)^2=25\pi\text{
cm}^2\).

To determine the circumference of the larger circle, we will first
find its radius.

Let \(r\) be the radius of the
larger circle. Then the area of the larger circle is \(\pi r^2\text{ cm}^2\).

Since the eight congruent pieces each have the same area as the
smaller circle, it follows that this area is equal to \(\frac{1}{9}\) of the area of the larger
circle. Thus, \[\begin{aligned}
25\pi &= \frac{1}{9} \pi r^2\\
25 &= \frac{1}{9} r^2\\
9 \times 25 &= r^2\\
225 &= r^2
\end{aligned}\] Thus, \(r=15\),
since \(r>0\).

It follows that the circumference of the larger circle is \(2\pi \times r = 2\pi \times 15 = 30\pi \text{
cm}\).