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?

The correct answer is √(3√3):2:√π.

Let us denote the common area of the equilateral triangle, the square, and the circle as $A$.

1. Equilateral Triangle:
The area of an equilateral triangle with side length $a$ is given by the formula:

$$A=\frac{\sqrt{3}}{4}{a}^2$$

Solving for the side length $a$ in terms of $A$:

$$a^2=\frac{4A}{\sqrt{3}}\Rightarrow a=\sqrt{\frac{4A}{\sqrt{3}}}=\frac{2\sqrt{A}}{\sqrt[4]{3}}$$

The perimeter of the equilateral triangle ($P_T$) is:

$$P_T=3a=3·\frac{2\sqrt{A}}{\sqrt[4]{3}}=2·3^{1-\frac{1}{4}}\sqrt{A}=2·3^{\frac{3}{4}}\sqrt{A}=2\sqrt{3\sqrt{3}}\sqrt{A}$$

2. Square:
The area of a square with side length $s$ is:

$$A=s^2\Rightarrow s=\sqrt{A}$$

The perimeter of the square ($P_S$) is:

$$P_S=4s=4\sqrt{A}$$

3. Circle:
The area of a circle with radius $r$ is:

$$A=\pi r^2\Rightarrow r=\sqrt{\frac{A}{\pi }}$$

The perimeter (circumference) of the circle ($P_C$) is:

$$P_C=2\pi r=2\pi \sqrt{\frac{A}{\pi }}=2\sqrt{\pi }\sqrt{A}$$

4. Ratio of the Perimeters:
We want to find the ratio of the perimeter of the equilateral triangle to the square to the circle ($P_T:P_S:P_C$):

$$P_T:P_S:P_C=2\sqrt{3\sqrt{3}}\sqrt{A}:4\sqrt{A}:2\sqrt{\pi }\sqrt{A}$$

Dividing all three terms by the common factor $2\sqrt{A}$ simplifies the ratio to:

$$P_T:P_S:P_C=\sqrt{3\sqrt{3}}:2:\sqrt{\pi }$$