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In the below figure, $ A B C D $ is a square of side $ 2 a $. Find the ratio between the areas of the incircle and the circum-circle of the square."

Academic Mathematics NCERT Class 10

Given:

\( A B C D \) is a square of side \( 2 a \).

To do:

We have to find the ratio between the areas of the incircle and the circum-circle of the square.

Solution:

The square $ABCD$ is inscribed a circle.
Length of the side of the square $= 2a$

From the figure,

Diameter of the outer circle $AC =$ Diagonal of the square

$=\sqrt{2} \times 2 a$

$=2 \sqrt{2} a$

This implies,

Radius of the outer circle $R=\frac{\mathrm{AC}}{2}$

$=\frac{2 \sqrt{2} a}{2}$

$=\sqrt{2} a$

Diameter of the inner circle $=2a$

Radius of the inner circle $r=\frac{2a}{2}=a$

Therefore,

The ratio between the areas of the incircle and the circum-circle of the square $= \frac{\text { Area of incircle }}{\text { Area of circumcircle }}$

$=\frac{\pi r^{2}}{\pi \mathrm{R}^{2}}$

$=\frac{\pi(\sqrt{2} a)^{2}}{\pi(a)^{2}}$

$=\frac{\pi 2 a^{2}}{\pi a^{2}}$

$=\frac{2}{1}$

The ratio between the areas of the incircle and the circum-circle of the square is $2:1$.

Updated on: 10-Oct-2022

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