Question

The ratio of the circumference of two circles is $3:1$. If the circumference of bigger circle is $39.36\mathrm{cm}$, find the radius of the smaller circle.

Answer 1

Step 1: Given information.

The ratio of the circumference of the two circles$=3:1$

The circumference of the bigger circle$=39.36\mathrm{cm}$

Step 2: Find the radius of the smaller circle.

Let the radius of the bigger circle and smaller circle be $r_1$ and $r_2$ respectively.

The circumference of of a circle with radius $r$ is,

$C=2\mathrm{\pi }r$

Let the circumference of the bigger circle and smaller circle be $C_1$ and $C_2$ respectively.

Then, the ratio of their circumference is,

$\begin{array}{rcl}\frac{C_1}{C_2}& =& \frac{3}{1}\\ & \Rightarrow & \frac{2\mathrm{\pi }{r}_1}{2\mathrm{\pi }{r}_2}=\frac{3}{1}\\ & \Rightarrow & \frac{39.36}{2\times \mathrm{\pi }\times r_2}=\frac{3}{1}\\ & \Rightarrow & r_2=\frac{39.36}{2\times {\displaystyle \mathrm{\pi }\times 3}}\\ & \Rightarrow & r_2=\frac{39.36}{6\times {\displaystyle \mathrm{\pi }}}\\ \therefore r_2& =& \frac{6.56}{{\displaystyle \mathrm{\pi }}}\mathrm{cm}\end{array}$

Thus, the radius of the smaller circle is $\frac{6.56}{{\displaystyle \mathrm{\pi }}}\mathrm{cm}$.