Question

If the circumference of a circle and the perimeter of a square are equal, then
(a) area of the circle = area of the square
(b) (area of the circle) > (area of the square)
(c) (area of the circle) < (area of the square)
(d) none of these

Answer 1

(b) Area of the circle > Area of the square
Let r be the radius of the circle.
We know:
Circumference of the circle$=2\mathrm{\pi }r$
Now,
Let a be the side of the square.
We know:
Perimeter of the square = 4a
Now,
$$\begin{gathered} 2\mathrm{\pi }r=4a \\ \Rightarrow r=\frac{4a}{2\mathrm{\pi }} \end{gathered}$$
∴ Area of the circle$=\mathrm{\pi }{r}^2$
$$\begin{gathered} =\mathrm{\pi }\times {\left(\frac{4a}{2\mathrm{\pi }}\right)}^2 \\ =\mathrm{\pi }\times \frac{16a^{\mathit{2}}}{4{\mathrm{\pi }}^2} \\ =\frac{4a^2}{\mathrm{\pi }} \\ =\frac{4\times 7a^2}{22} \\ =\frac{14a^2}{11} \end{gathered}$$

Also,
Area of the square$=a^2$
Clearly, $\frac{14a^2}{11}>a$².
∴ Area of the circle > Area of the square