The circumference of a circle is 10% more than the perimeter of a square. If the difference between the area of the circle and that of the square is 216 cm2, how much does the diagonal of the square measure ?

Answer: Option D

Let the radius of circle be r cm and side of square be a cm
Then circumference of circle = $$2\pi r$$ and
Perimeter of square = 4a
According to the question,
$$\begin{array}{rl}& 2\pi r=4a\times \frac{110}{100}\\ & \Rightarrow 2\pi r=\frac{44a}{10}\\ & \Rightarrow r=\frac{44a}{2\pi \times 10}\\ & \Rightarrow r=\frac{11a}{5\pi }\\ & \Rightarrow a=\frac{5\pi r}{11}.....(i)\\ & \text{Also, }\pi r^2-a^2\Rightarrow 216\end{array}$$
$$\Rightarrow \pi r^2-\frac{25\pi r^2}{121}=216$$     $$\left[\text{from equation (i)}\right]$$
$$\begin{array}{rl}& \Rightarrow \frac{121\pi r^2-25\pi r^2}{121}=216\\ & \Rightarrow r^2\left[121\pi -25{\pi }^2\right]=26136\end{array}$$
$$\Rightarrow r^2\left[121\times \frac{22}{7}-25\times \frac{22}{7}\times \frac{22}{7}\right]$$      $$=26136$$
$$\begin{array}{rl}& \Rightarrow r^2\left[\frac{2662}{7}-\frac{12100}{49}\right]=26136\\ & \Rightarrow r^2\left[\frac{6534}{49}\right]=26136\\ & \Rightarrow r^2=\frac{26136\times 49}{6534}\\ & \Rightarrow r^2=196\\ & \Rightarrow r=14cm\\ & \therefore a=\frac{5\pi r}{11}=5\times \frac{22}{7}\times \frac{14}{11}=20cm\end{array}$$
Hence, diagonal of square $$=\sqrt{2}a=20\sqrt{2}cm$$