Question

If a circle is inscribed in a square, then

• A. $\frac{\text{Area of the circle}}{\text{Area of the square}}=\frac{\pi }{2}$
• B. $\frac{\text{Area of the circle}}{\text{Area of the square}}=\frac{\pi }{4}$
• C. $\frac{\text{Circumference of the circle}}{\text{Perimeter of the square}}=\frac{\pi }{4}$
• D. $\frac{\text{Circumference of the circle}}{\text{Perimeter of the square}}=\frac{\pi }{2}$

Answer 1

Answer: (B), (C)

$\frac{\text{Circumference of the circle}}{\text{Perimeter of the square}}=\frac{\pi }{4}$


Let the radius of the circle and the side length of the square be $r$ and $s$ respectively.

Step 1: Area of the circle $=\pi r^2$
Circumference of the circle $=2\pi r$

Step 2: Diameter of the circle $=2\times r$

Step 3: When a circle is inscribed in a square, the diameter of the circle is equal to the side length of the square.


So, the side length of the square$\left(s\right)$ is $2r.$

Step 4:
$\text{Area of the square}=s^2=(2r{)}^2=2^2\times r^2=4r^2$

$\text{Perimeter of the square}=4\times s=4\times 2r=8r$

Step 5:
$\text{Area of the circle : Area of the square}=\pi r^2:4r^2$

Divide both sides by $r^2$
$=\frac{\pi r^2}{r^2}:\frac{4r^2}{r^2}$
$=\pi :4$

$\text{Circumference of the circle : Perimeter of the square}$
$=2\pi r:8r$
Divide both sides by $2r$
$=\frac{2\pi r}{2r}:\frac{8r}{2r}=\pi :4$

Hence, options (b) and (c) are correct choices.