Question

The perimeter of a square and a regular hexagon are equal. The ratio of the area of the hexagon the area of the square is :

• A. \(2\sqrt3:1\)
• B. \(2:\sqrt3\)
• C. \(3\sqrt3:2\)
• D. \(\sqrt2:3\)

Answer 1

The Correct Option is B

To find the ratio of the area of a regular hexagon to the area of a square, given that their perimeters are equal, follow these steps:

1. Let the side length of the square be \( s \). Thus, the perimeter of the square is \( 4s \).

2. Let the side length of the regular hexagon be \( a \). Thus, the perimeter of the hexagon is \( 6a \).

3. Since the perimeters are equal: \( 4s = 6a \). Solving for \( s \) in terms of \( a \):

$$ s = \frac{3}{2}a $$

4. The area of the square is \( s^2 \):

$$ s^2 = \left(\frac{3}{2}a\right)^2 = \frac{9}{4}a^2 $$

5. The area of the regular hexagon is given by:

$$ \text{Area of Hexagon} = \frac{3\sqrt{3}}{2}a^2 $$

6. The ratio of the area of the hexagon to the square is:

$$ \text{Ratio} = \frac{\frac{3\sqrt{3}}{2}a^2}{\frac{9}{4}a^2} $$

7. Simplifying the ratio:

$$ = \frac{3\sqrt{3}}{2} \times \frac{4}{9} = \frac{3\sqrt{3} \times 4}{2 \times 9} = \frac{12\sqrt{3}}{18} $$

8. Further simplification gives:

$$ = \frac{2\sqrt{3}}{3} $$

9. Therefore, the ratio of the area of the hexagon to that of the square is \( \frac{2}{\sqrt{3}} \) or simply:

Option: \(2:\sqrt{3}\).