The side of a square is equal to the side of an equilateral triangle ,The ratio of their areas is

Options

A. `4:3`
B. `2: sqrt(3)`
C. `4:sqrt(3)`
D. None of these

Explanation

To find the ratio of the areas of a square and an equilateral triangle when both have the same side length, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Side Length**:
Let the side length of both the square and the equilateral triangle be denoted as \( s \).

2. **Calculate the Area of the Square**:
The area \( A_{square} \) of a square is given by the formula:
\[ A_{square} = s^2 \]

3. **Calculate the Area of the Equilateral Triangle**:
The area \( A_{triangle} \) of an equilateral triangle is given by the formula:
\[ A_{triangle} = \frac{\sqrt{3}}{4} s^2 \]

4. **Set Up the Ratio of the Areas**:
Now, we need to find the ratio of the area of the square to the area of the equilateral triangle:
\[ \text{Ratio} = \frac{A_{square}}{A_{triangle}} = \frac{s^2}{\frac{\sqrt{3}}{4} s^2} \]

5. **Simplify the Ratio**:
Since \( s^2 \) is common in both the numerator and the denominator, we can cancel it out:
\[ \text{Ratio} = \frac{1}{\frac{\sqrt{3}}{4}} = \frac{4}{\sqrt{3}} \]

6. **Rationalize the Denominator** (optional):
To express the ratio in a more standard form, we can rationalize the denominator:
\[ \text{Ratio} = \frac{4}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3} \]

### Final Answer:
The ratio of the areas of the square to the equilateral triangle is:
\[ \frac{4\sqrt{3}}{3} \]