Question

By bending a wire, a square is made whose area is x sq. units. If a circle is made from same wire, what will be its area? Which of the two is greater in area?

Answer 1

Area of the square = x sq units
$\Rightarrow$ (side)² = x
$\Rightarrow \mathrm{side}=\sqrt{x}$

Perimeter of the square =Circumference of the circle
$\Rightarrow$ 4 × side = 2$\mathrm{\pi }$ × radius
$$\begin{gathered} \Rightarrow 4\sqrt{x}=2\mathrm{\pi }\times \mathrm{radius} \\ \Rightarrow \mathrm{radius}=\frac{4\sqrt{x}}{2\mathrm{\pi }}=\frac{2\sqrt{x}}{\mathrm{\pi }} \end{gathered}$$

$$\begin{gathered} \mathrm{Now},\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{circle}=\pi {\left(\mathrm{radius}\right)}^2 \\ =\pi {\left(\frac{2\sqrt{x}}{\mathrm{\pi }}\right)}^2 \\ =\frac{4x\times 7}{22} \\ =\frac{14}{11}x\mathit{} \end{gathered}$$

$\therefore$ Area of the circle is greater than that of the square.