Answer :

Solution :

Let the side of an equilateral triangle, side of a square, the radius of a circle be 𝑎,𝑥, and 𝑟 respectively.

Now, $\frac{\sqrt{3}}{4}{a}^2=x^2=\pi r^2=k^2(\text{let})$

Now,

$\frac{\sqrt{3}}{4}{a}^2=k^2\Rightarrow a^2=\frac{4k^2}{\sqrt{3}}\Rightarrow \boxed{{\displaystyle a=\frac{2k}{\sqrt{\sqrt{3}}}}}$

$x^2=k^2\Rightarrow \boxed{{\displaystyle x=k}}$

$\pi r^2=k^2\Rightarrow \boxed{{\displaystyle r=\frac{k}{\sqrt{\pi }}}}$

Now, we can calculate the perimeter of each of that.

The perimeter of an equilateral triangle = 3𝑎

The perimeter of a square = 4𝑥

The perimeter of a circle = 2Πr

The ratio of the perimeters of the equilateral triangle to square to circle = 3a : 4x : 2Πr

$=3\times {\displaystyle \frac{2k}{\sqrt{\sqrt{3}}}}:4k:2\pi \times {\displaystyle \frac{k}{\sqrt{\pi }}}$

$={\displaystyle \frac{3}{\sqrt{\sqrt{3}}}}\times {\displaystyle \frac{\sqrt{\sqrt{3}}}{\sqrt{\sqrt{3}}}}:2:\sqrt{{\displaystyle \frac{{\pi }^2}{\pi }}}$

$={\displaystyle \frac{3\sqrt{\sqrt{3}}}{\sqrt{3}}}:2:\sqrt{\pi }$

$={\displaystyle \frac{3\sqrt{\sqrt{3}}}{\sqrt{3}}}\times {\displaystyle \frac{\sqrt{3}}{\sqrt{3}}}:2:\sqrt{\pi }$

$=\sqrt{(3\sqrt{3})}:2:\sqrt{\pi }$