If the side of an equilateral triangle is r, then the area of the triangle varies directly as .

Options

A. `sqrt(r)`
B. `r`
C. `r^(2) `
D. `r^(3) `

Answer

To solve the problem, we need to determine how the area of an equilateral triangle varies with respect to the length of its side, denoted as \( r \).

### Step-by-Step Solution:

1. **Understand the formula for the area of an equilateral triangle**:
The area \( A \) of an equilateral triangle with side length \( r \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} r^2 \]

2. **Identify the relationship**:
From the area formula, we can see that the area \( A \) is proportional to the square of the side length \( r \). This means that if we change the side length \( r \), the area will change according to the square of that change.

3. **Express the proportionality**:
We can express this relationship mathematically as:
\[ A \propto r^2 \]
This indicates that the area \( A \) varies directly as \( r^2 \).

4. **Conclusion**:
Therefore, we conclude that the area of the equilateral triangle varies directly as \( r^2 \).

### Final Answer:
The area of the triangle varies directly as \( r^2 \).

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