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If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.
Given:
A square is inscribed in a circle.
To do:
We have to find the ratio of the areas of the circle and the square.
Solution:
Let $r$ be the radius of the circle $s$ be the side of the square.
This implies,
$AB = BC = CD = DA = s$
$AC$ and $BD$ are the diagonals of the square.
Diagonal $AC$ of the square $=$ Diameter of the circle
$\Rightarrow \sqrt{2} s=2 r$
$\Rightarrow s=\frac{2 r}{\sqrt{2}}$
Area of the circle $=\pi r^{2}$
Area of the square $=s^{2}$
The ratio of the areas of the circle and the square $=\frac{\pi r^{2}}{s^{2}}$
$=\frac{22 r^{2}}{7 \times(\frac{2 r}{\sqrt{2}})^{2}}$
$=\frac{22 r^{2}}{7 \times \frac{4 r^{2}}{2}}$
$=\frac{22 r^{2} \times 2}{7 \times 4 r^{2}}$
$=\frac{11}{7}$
The ratio of the areas of the circle and the square is $11:7$.