The height of an equilateral triangle is `3sqrt(3) ` cm . Its area is

Options

A. `6sqrt(3) cm^(2)`
B. `27 cm^(2)`
C. `9sqrt(3) cm^(2)`
D. `27 sqrt( 3) cm^(2)`

Explanation

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

### Step 1: Understand the relationship between height and side length
For an equilateral triangle, the height (h) can be expressed in terms of the side length (s) using the formula:
\[ h = \frac{\sqrt{3}}{2} s \]

### Step 2: Substitute the given height into the formula
We know the height of the triangle is \( 3\sqrt{3} \) cm. We can set up the equation:
\[ 3\sqrt{3} = \frac{\sqrt{3}}{2} s \]

### Step 3: Solve for the side length (s)
To isolate \( s \), we can multiply both sides of the equation by 2:
\[ 2 \times 3\sqrt{3} = \sqrt{3} s \]
\[ 6\sqrt{3} = \sqrt{3} s \]

Now, divide both sides by \( \sqrt{3} \):
\[ s = 6 \text{ cm} \]

### Step 4: Calculate the area using the side length
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]

Substituting \( s = 6 \) cm into the area formula:
\[ A = \frac{\sqrt{3}}{4} (6)^2 \]
\[ A = \frac{\sqrt{3}}{4} \times 36 \]
\[ A = 9\sqrt{3} \text{ cm}^2 \]

### Final Answer
The area of the equilateral triangle is \( 9\sqrt{3} \text{ cm}^2 \).

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