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 ?

Solution(By Examveda Team)

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,
$$\eqalign{ & 2\pi r = 4a \times \frac{{110}}{{100}} \cr & \Rightarrow 2\pi r = \frac{{44a}}{{10}} \cr & \Rightarrow r = \frac{{44a}}{{2\pi \times 10}} \cr & \Rightarrow r = \frac{{11a}}{{5\pi }} \cr & \Rightarrow a = \frac{{5\pi r}}{{11 }} . . . . .(i) \cr & {\text{Also, }}\pi {r^2} - {a^2} \Rightarrow 216 \cr} $$
$$ \Rightarrow \pi {r^2} - \frac{{25\pi {r^2}}}{{121}} = 216$$     $$\left[ {{\text{from equation (i)}}} \right]$$
$$\eqalign{ & \Rightarrow \frac{{121\pi {r^2} - 25\pi {r^2}}}{{121}} = 216 \cr & \Rightarrow {r^2}\left[ {121\pi - 25{\pi ^2}} \right] = 26136 \cr} $$
$$ \Rightarrow {r^2}\left[ {121 \times \frac{{22}}{7} - 25 \times \frac{{22}}{7} \times \frac{{22}}{7}} \right]$$      $$ = 26136$$
$$\eqalign{ & \Rightarrow {r^2}\left[ {\frac{{2662}}{7} - \frac{{12100}}{{49}}} \right] = 26136 \cr & \Rightarrow {r^2}\left[ {\frac{{6534}}{{49}}} \right] = 26136 \cr & \Rightarrow {r^2} = \frac{{26136 \times 49}}{{6534}} \cr & \Rightarrow {r^2} = 196 \cr & \Rightarrow r = 14\,cm \cr & \therefore a = \frac{{5\pi r}}{{11}} = 5 \times \frac{{22}}{7} \times \frac{{14}}{{11}} = 20\,cm \cr} $$
Hence, diagonal of square $$ = \sqrt 2 a = 20\sqrt 2 \,cm$$