Question

The area of circle is 100 times the area of another circle. What is the ratio of their circumferences?

Answer 1

Let the area of the first circle be A₁, the circumference be C₁ and the radius be r₁.
Let the area of the second circle be A_{2 ,} the circumference be C₂ and the radius be r₂.

Thus,

$$\begin{gathered} C_1:C_2=2\mathrm{\pi }{r}_1:2\mathrm{\pi }{r}_2 \\ \Rightarrow \frac{C_1}{C_2}=\frac{2\mathrm{\pi }{r}_1}{2{\mathrm{\pi r}}_2}=\frac{r_1}{r_2} \end{gathered}$$

We know that :

A₁ = 100A_{2
∴ $$\begin{gathered} \Rightarrow \left({{\mathrm{\pi r}}_1}^2\right)=100\times \left({{\mathrm{\pi r}}_2}^2\right) \\ \Rightarrow {{\mathrm{r}}_1}^2=100\times {{\mathrm{r}}_2}^2 \\ \Rightarrow {\mathrm{r}}_1=10\times {\mathrm{r}}_2 \\ \Rightarrow \frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}=10 \end{gathered}$$}

Substituting the values, we get:
∴​ $$\begin{gathered} \frac{C_1}{C_2}=\frac{r_1}{r_2}=\frac{10}{1} \\ {C}_1:C_2=10:1 \end{gathered}$$

Hence, the ratio of their circumferences is 10:1.