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The side of a square is 10 cm. Find the area of circumscribed and inscribed circles.
Given:
The side of a square is 10 cm.
To do:
We have to find the area of circumscribed and inscribed circles.
Solution:
Let $ABCD$ be a square whose each side is $10\ cm$, $r_1$ be the radius of the circumscribed circle and $r_2$ be the radius of the inscribed circle.
This implies,
$AB = BC = CD = DA = 10\ cm$
$AC$ and $BD$ are the diagonals of the square.
$\mathrm{AC}=\mathrm{BD}=\sqrt{2}\times10$
$=1.414 \times 10$
$=14.14 \mathrm{~cm}$
Radius of the circumscribed circle $\mathrm{r_1}=\frac{\mathrm{AC}}{2}$
$=\frac{14.1}{2}$
$=7.05 \mathrm{~cm}$
Radius of the inscribed circle $r_2=\frac{\mathrm{AB}}{2}$
$=\frac{10}{2}$
$=5 \mathrm{~cm}$
Area of the circumscribed circle $=\pi \mathrm{r_1}^{2}$
$=\frac{22}{7} \times(7.07)^{2}$
$=\frac{22}{7} \times 7.07 \times 7.07$
$=22 \times 1.01 \times 7.07$
$=157.0954$
$=157.1 \mathrm{~cm}^{2}$
Area of the inscribed circle $=\pi r_2^{2}$
$=\frac{22}{7} \times 5 \times 5 \mathrm{~cm}^{2}$
$=\frac{550}{7}$
$=78.57 \mathrm{~cm}^{2}$
The area of the circumscribed and the inscribed circles is $157.1\ cm^2$ and $78.57\ cm^2$ respectively.