Mathematico | Mathematico

The circles below have the property that the area of the smaller circle is equal to the circumference of the larger circle.

Circle within circle

Find the radius of the smaller circle, giving your solution to $3$ significant figures.

Hint

Let the smaller circle have radius $r$ and the larger circle have radius $R$ .

Then

2 R = 2 r + 1

Hint

The relationship between the two circles can be expressed as

π r^{2} = 2 π R

(It will help to cancel $π$ before continuing.)

Solution

Let the smaller circle have radius $r$ and the larger circle have radius $R$ .

Then

2 R = 2 r + 1

The relationship between the two circles can be expressed as

\begin{aligned} π r^{2} & = 2 π R \\ r^{2} & = 2 R \\ r^{2} & = 2 r + 1 \\ r^{2} - 2 r - 1 & = 0 \end{aligned}

so by the quadratic formula

\begin{aligned} r & = \frac{2 \pm \sqrt{4 - 4 (1) (- 1)}}{2} \\ = 1 \pm \sqrt{2} \end{aligned}

Clearly $r > 0$ , so $r = 1 + \sqrt{2} = 2.41 (3 s.f.)$ is the only solution.