If the height of the equilateral triangle is `3` cm, then determine the area (in cm2) of the equilateral triangle.

Options

A. 6
B. `2sqrt3`
C. `3sqrt3`
D. 12

Answer

To find the area of an equilateral triangle when given its height, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the height of the equilateral triangle**:
The problem states that the height (h) of the equilateral triangle is 3 cm.

2. **Use the formula for the height of an equilateral triangle**:
The height (h) of an equilateral triangle can be expressed in terms of its side length (a) as:
\[ h = \frac{\sqrt{3}}{2} a \]

3. **Substitute the height into the formula**:
We know that \( h = 3 \) cm, so we can set up the equation:
\[ 3 = \frac{\sqrt{3}}{2} a \]

4. **Solve for the side length (a)**:
To isolate \( a \), multiply both sides of the equation by 2:
\[ 6 = \sqrt{3} a \]
Now, divide both sides by \( \sqrt{3} \):
\[ a = \frac{6}{\sqrt{3}} = 2\sqrt{3} \text{ cm} \]

5. **Use the formula for the area of an equilateral triangle**:
The area (A) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

6. **Substitute the value of a into the area formula**:
Now substitute \( a = 2\sqrt{3} \) into the area formula:
\[ A = \frac{\sqrt{3}}{4} (2\sqrt{3})^2 \]

7. **Calculate \( (2\sqrt{3})^2 \)**:
\[ (2\sqrt{3})^2 = 4 \cdot 3 = 12 \]

8. **Substitute back into the area formula**:
\[ A = \frac{\sqrt{3}}{4} \cdot 12 \]

9. **Simplify the area**:
\[ A = 3\sqrt{3} \text{ cm}^2 \]

### Final Answer:
The area of the equilateral triangle is \( 3\sqrt{3} \text{ cm}^2 \).

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