Question

The side of a square is 10 cm. Find (i) the area of the inscribed circle, and (ii) the area of the circumscribed circle.

Answer 1

(​i)​ If a circle is inscribed in a square, then the side of the square is equal to the diameter of the circle.
Side of the square = 10 cm
Side = Diameter = 10
∴ Radius = 5 cm
Area of the inscribed circle = ${\mathrm{\pi r}}^2$
$$\begin{gathered} =3.14\times 5\times 5 \\ =78.5{\mathrm{cm}}^2 \end{gathered}$$

(ii) If a circle is circumscribed in a square, then the diagonal of the square is equal to the diameter of the circle.
$$\begin{gathered} \mathrm{Diagonal}\mathrm{of}\mathrm{the}\mathrm{square}=\sqrt{2}\times \mathrm{Side}\mathrm{of}\mathrm{the}\mathrm{square} \\ =\sqrt{2}\times 10 \\ =10\sqrt{2}\mathrm{cm} \end{gathered}$$

Diagonal = Diameter = $10\sqrt{2}\mathrm{cm}$
∴ $r=5\sqrt{2}$ cm

Now,
Area of the circumscribed circle = ${\mathrm{\pi r}}^2$
$$\begin{gathered} =3.14\times {\left(5\sqrt{2}\right)}^2 \\ =3.14\times 50 \\ =157{\mathrm{cm}}^2 \end{gathered}$$