Question

If the perimeter of a square is equal to the circumference of a circle then the ratio of their areas is
(a) 4 : π
(b) π : 4
(c) π : 7
(d) 7 : π

Answer 1

Let the side of the square be a and the radius of the circle be r.
Now, Perimeter of circle = Circumference of the circle
$$\begin{gathered} \Rightarrow 4a=2\mathrm{\pi }r \\ \Rightarrow \frac{a}{r}=\frac{\mathrm{\pi }}{2} \\ \mathrm{Now}, \\ \frac{\mathrm{Area}\mathrm{of}\mathrm{square}}{\mathrm{Area}\mathrm{of}\mathrm{circle}}=\frac{a^2}{\mathrm{\pi }{r}^2} \\ =\frac{1}{\mathrm{\pi }}\times {\left(\frac{a}{r}\right)}^2 \\ =\frac{1}{\mathrm{\pi }}\times \frac{{\mathrm{\pi }}^2}{4} \\ =\frac{\mathrm{\pi }}{4} \end{gathered}$$
Hence, the correct answer is option (b).