Question

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?

• A. \( 3\sqrt{3} : 2 : \sqrt{\pi} \)
• B. \( \sqrt{3\sqrt{3}} : 2 : \sqrt{\pi} \)
• C. \( \sqrt{3\sqrt{3}} : 4 : 2\sqrt{\pi} \)
• D. \( \sqrt{3\sqrt{3}} : 2 : 2\sqrt{\pi} \)

Hint:

For geometric figures with equal areas, calculate the perimeter using their respective formulas and then compute the ratio of the perimeters.

Answer 1

The Correct Option is B

Let the area of the equilateral triangle, square, and circle be denoted by \( A \). Since they all have equal areas, we can calculate their perimeters based on their area formulas.
Step 1: Area of the equilateral triangle The area of an equilateral triangle with side \( s \) is given by: \[ A = \frac{s^2\sqrt{3}}{4}. \] From this, we can express the side \( s \) in terms of the area \( A \): \[ s = \sqrt{\frac{4A}{\sqrt{3}}}. \] The perimeter of the equilateral triangle is \( 3s \), so the perimeter is: \[ P_{\text{triangle}} = 3 \times \sqrt{\frac{4A}{\sqrt{3}}}. \]
Step 2: Area of the square The area of the square with side \( a \) is: \[ A = a^2. \] The perimeter of the square is: \[ P_{\text{square}} = 4a = 4\sqrt{A}. \]
Step 3: Area of the circle The area of the circle with radius \( r \) is: \[ A = \pi r^2. \] The perimeter (circumference) of the circle is: \[ P_{\text{circle}} = 2\pi r = 2\sqrt{\frac{A}{\pi}}. \]
Step 4: Ratio of perimeters We now find the ratio of the perimeters: \[ \frac{P_{\text{triangle}}}{P_{\text{square}}} = \frac{3\sqrt{\frac{4A}{\sqrt{3}}}}{4\sqrt{A}} = \sqrt{3\sqrt{3}}, \] \[ \frac{P_{\text{square}}}{P_{\text{circle}}} = \frac{4\sqrt{A}}{2\sqrt{\frac{A}{\pi}}} = \sqrt{\pi}. \]
Thus, the ratio of the perimeters of the equilateral triangle to the square to the circle is: \[ \sqrt{3\sqrt{3}} : 2 : \sqrt{\pi}. \] The correct answer is (B) \( \sqrt{3\sqrt{3}} : 2 : \sqrt{\pi} \).