A square and a regular hexagon have equal perimeters. Their areas are in the ratio:

Options

A. `2:1`
B. `2sqrt(3):1`
C. `sqrt(3):2`
D. `3:2`

Answer

To solve the problem of finding the ratio of the areas of a square and a regular hexagon with equal perimeters, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Perimeter Equality**:
- Let the side length of the square be \( a \).
- The perimeter of the square is \( P_{\text{square}} = 4a \).
- Let the side length of the regular hexagon be \( b \).
- The perimeter of the hexagon is \( P_{\text{hexagon}} = 6b \).
- Since the perimeters are equal, we have:
\[ 4a = 6b \]

2. **Expressing \( b \) in terms of \( a \)**:
- Rearranging the equation \( 4a = 6b \):
\[ b = \frac{2}{3}a \]

3. **Calculating the Area of the Square**:
- The area of the square \( A_{\text{square}} \) is given by:
\[ A_{\text{square}} = a^2 \]

4. **Calculating the Area of the Regular Hexagon**:
- The area \( A_{\text{hexagon}} \) of a regular hexagon can be calculated using the formula:
\[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2}b^2 \]
- Substituting \( b = \frac{2}{3}a \) into the area formula:
\[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2}\left(\frac{2}{3}a\right)^2 \]
- Simplifying this:
\[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \cdot \frac{4}{9}a^2 = \frac{2\sqrt{3}}{3}a^2 \]

5. **Finding the Ratio of Areas**:
- Now, we need to find the ratio of the area of the square to the area of the hexagon:
\[ \text{Ratio} = \frac{A_{\text{square}}}{A_{\text{hexagon}}} = \frac{a^2}{\frac{2\sqrt{3}}{3}a^2} \]
- The \( a^2 \) terms cancel out:
\[ \text{Ratio} = \frac{1}{\frac{2\sqrt{3}}{3}} = \frac{3}{2\sqrt{3}} \]
- To rationalize the denominator:
\[ \text{Ratio} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2} \]

### Final Answer:
The ratio of the areas of the square and the regular hexagon is \( \frac{\sqrt{3}}{2} \).

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