Question

A wire of 5024 m length is in the form of a square. It is cut and made a circle. Find the ratio of the area of the square to that of the circle.

Answer 1

We have:
Perimeter of the square = 5024 m = Circumference of the circle
⇒​ 4 x Side of the square = 5024
∴ Side of the square = $\frac{5024}{4}=1256\mathrm{m}$.
Let the area of the square be A₁ and the area of the circle be A₂.
Area of the square (A₁)= side x side =$\left(\frac{5024}{4}\times \frac{5024}{4}\right){\mathrm{m}}^2$ .
Circumference of the circle = 5024 m
⇒ 2 $\mathrm{\pi }$r = 5024 m

$$\begin{gathered} \Rightarrow \left(2\times \frac{22}{7}\times r\right)=5024\mathrm{m} \\ \Rightarrow r=\frac{5024\times 7}{2\times 22}. \end{gathered}$$

Area of the circle (A₂)= $\mathrm{\pi }$r² = ​$\left[\frac{22}{7}\times \frac{5024\times 7}{2\times 22}\times \frac{5024\times 7}{2\times 22}\right]=\left[\frac{5024\times 5024\times \times 7}{2\times 2\times 22}\right]{\mathrm{m}}^2$.

$$\begin{gathered} \therefore A_1:A_2=\left[\frac{5024}{4}\times \frac{5024}{4}\right]:\left[\frac{5024\times 5024\times \times 7}{2\times 2\times 22}\right] \\ \Rightarrow \frac{A_1}{A_2}=\left[\frac{5024}{4}\times \frac{5024}{4}\right]÷\left[\frac{5024\times 5024\times \times 7}{2\times 2\times 22}\right] \\ \Rightarrow \frac{A_1}{A_2}=\left[\frac{5024}{4}\times \frac{5024}{4}\right]\times \left[\frac{2\times 2\times 22}{5024\times 5024\times \times 7}\right] \\ \Rightarrow \frac{A_1}{A_2}=\frac{11}{14}. \\ \Rightarrow A_1:A_2=11:14. \end{gathered}$$
Hence, the ratio of the area of the square to the area of the circle is 11:14.