The ratio of the area of a square of side a and that of an equilateral triangle of side a, is

Options

A. `2:1`
B. `2:sqrt3`
C. `4:3`
D. `4:sqrt3`

Answer

To find the ratio of the area of a square with side length \( a \) and that of an equilateral triangle with side length \( a \), we can follow these steps:

### Step 1: Calculate the Area of the Square
The formula for the area of a square is given by:
\[ \text{Area of Square} = \text{side} \times \text{side} = a \times a = a^2 \]

### Step 2: Calculate the Area of the Equilateral Triangle
The formula for the area of an equilateral triangle is:
\[ \text{Area of Equilateral Triangle} = \frac{\sqrt{3}}{4} a^2 \]

### Step 3: Set Up the Ratio
Now, we need to find the ratio of the area of the square to the area of the equilateral triangle:
\[ \text{Ratio} = \frac{\text{Area of Square}}{\text{Area of Equilateral Triangle}} = \frac{a^2}{\frac{\sqrt{3}}{4} a^2} \]

### Step 4: Simplify the Ratio
We can simplify the ratio:
\[ \text{Ratio} = \frac{a^2}{\frac{\sqrt{3}}{4} a^2} = \frac{a^2 \cdot 4}{\sqrt{3} \cdot a^2} \]
Since \( a^2 \) in the numerator and denominator cancels out, we have:
\[ \text{Ratio} = \frac{4}{\sqrt{3}} \]

### Step 5: Express the Ratio in a Standard Form
To express this ratio in a more conventional format, we can write it as:
\[ \text{Ratio} = 4 : \sqrt{3} \]

### Conclusion
Thus, the ratio of the area of the square to the area of the equilateral triangle is:
\[ \text{Ratio} = 4 : \sqrt{3} \]

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