Question

In the figure given alongside, a circle is inscribed in a square of side 4cm and another circle is circumscribing the square. Prove that the area of the circumscribed circle is two times the area of the inscribed circle.

Answer 1

ABCD is a square of side 4 cm.
Let r and R be the radius of the inscribed and circumscribed circle respectively.
Diameter of the inscribed circle = 2r
$\therefore$ 2r = 4 cm
$\Rightarrow$ r = 2 cm
Diameter of the circumscribed circle = Length of diagonal of the square
$\therefore 2R=\sqrt{2}\times Sideofthesquare\Rightarrow 2R=4\sqrt{2}cm\Rightarrow R=2\sqrt{2}cm\frac{Areaofthecircumscribed}{Areaofinscribedcircle}=\frac{\pi R^2}{\pi r^2}={\left(\frac{R}{r}\right)}^2={\left(\frac{2\sqrt{2}}{2}\right)}^2=(\sqrt{2}{)}^2=2$
$\therefore$ Area of circumscribed circle $=2\times Areaofinscribedcircle$