Areas of two circles are equal. Is it necessary that their circum… - Vidyadip

Question

Areas of two circles are equal. Is it necessary that their circumferences are equal? Why?

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Answer

True:
$\because\ \ \text{A}_1=\text{A}_2$ [Given]
$\Rightarrow\ \ \pi\text{r}_1^2=\pi\text{r}_2^2$
$\Rightarrow\ \ \text{r}_1^2=\text{r}_2^2$
$\Rightarrow\ \ \text{r}_1=\text{r}_2=\text{r}$ (say) [taking square root]
Now, $\frac{\text{C}_1}{\text{C}_2}=\frac{2\pi\text{r}_1}{2\pi\text{r}_2}=\frac{\text{r}_1}{\text{r}_2}=\frac{\text{r}}{\text{r}}=\frac{1}{1}$
Hence, $C_1 = C_2$
Hence, the given statement is true.

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