Question

The area of a square is the same as the area of a square. Their perimeters are in the ratio

पर्याय

• 1 : 1

• 2 : π

• π : 2

• ${\displaystyle \sqrt{\pi }:2}$

Answer 1

${\displaystyle \sqrt{\pi }:2}$
Let a be the side of the square.
We know:
Area of a square = a² 
Let r be the radius of the circle.
We know:
Area of a circle${\displaystyle =\pi {\text{r}}^2}$

Because the area of the square is the same as the area of the circle, we have:

a2 = πr2

${\displaystyle \Rightarrow \frac{{\text{r}}^2}{{\text{a}}^2}=\frac{1}{\pi }}$

${\displaystyle \Rightarrow \frac{\text{r}}{\text{a}}=\frac{1}{\sqrt{\pi }}}$

∴ Ratio of their perimeters

${\displaystyle =\frac{2\pi \text{r}}{4\text{a}}[\text{Because perimeter of the circle is 2πr and permeter} \text{of the square is 4}a]}$

${\displaystyle =\frac{\pi \text{r}}{2\text{a}}}$

${\displaystyle =\frac{\pi }{2}\times \frac{\text{r}}{\text{a}}}$

${\displaystyle =\frac{\pi }{2}\times \frac{1}{\sqrt{\pi }} [\text{Since}\frac{\text{r}}{\text{a}}=\frac{1}{\sqrt{\pi }}]}$

${\displaystyle =\frac{\sqrt{\pi }}{2}}$

${\displaystyle =\sqrt{\pi }:2}$