An equilateral triangle, a square and a circle have equal areas.

What is the ratio of the perimeters of the equilateral triangle to square to circle?

1. $3\sqrt{3} : 2 : \sqrt{\pi}$
2. $\sqrt{\left ( 3 \sqrt{3} \right )} : 2 : \sqrt{\pi}$
3. $\sqrt{\left ( 3 \sqrt{3} \right )} : 4 : 2\sqrt{\pi}$
4. $\sqrt{\left ( 3 \sqrt{3} \right )} : 2 : 2\sqrt{\pi}$

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,

• $\frac{\sqrt{3}}{4}\; a^{2} = k^{2} \Rightarrow a^{2} = \frac{4k^{2}}{\sqrt{3}} \Rightarrow {\color{Blue}{\boxed{a = \frac{2k}{\sqrt{\sqrt{3}}}}}}$
• $x^{2} = k^{2} \Rightarrow {\color{Purple}{\boxed{x = k}}}$
• $\pi r^{2} = k^{2} \Rightarrow {\color{Teal}{\boxed{r = \frac{k}{\sqrt{\pi}}}}}$

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 $= 2\pi r$

The ratio of the perimeters of the equilateral triangle to square to circle $= 3a:4x:2\pi r$

$\qquad \qquad = 3 \times \dfrac{2k}{\sqrt{\sqrt{3}}} : 4k:2 \pi \times \dfrac{k}{\sqrt{\pi}}$

$\qquad \qquad = \dfrac{3}{\sqrt{\sqrt{3}}} \times \dfrac{\sqrt{\sqrt{3}}}{\sqrt{\sqrt{3}}} : 2: \sqrt{\dfrac{\pi ^{2}}{ \pi}}$

$\qquad \qquad = \dfrac{3\;\sqrt{\sqrt{3}}}{\sqrt{3}} : 2: \sqrt{\pi}$

$\qquad \qquad = \dfrac{3\;\sqrt{\sqrt{3}}}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} : 2: \sqrt{\pi}$

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

Correct Answer $:\text{B}$

$\textbf{Important Points:}$

• The area of an equilateral triangle ${\color{Green}{ = \dfrac{\sqrt{3}}{4}\;(\text{Side of an equilateral triangle})^{2}}}$

• The area of a square ${\color{Lime}{ = (\text{Side of a square})^{2}}}$

• The area of a circle ${\color{Cyan}{= (\text{Radius})^{2}}}$

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