If the area of an equilateral triangle is `81 sqrt(3) cm^(2)`, find its height.

Options

A. `9 sqrt(3) cm`
B. `10 sqrt(3) cm`
C. `8 sqrt(3) cm`
D. `11 sqrt(3) cm`

Explanation

To find the height of an equilateral triangle with an area of \( 81 \sqrt{3} \, \text{cm}^2 \), we can follow these steps:

### Step 1: Use the formula for the area of an equilateral triangle.
The area \( A \) of an equilateral triangle with side length \( a \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Given that the area \( A = 81 \sqrt{3} \, \text{cm}^2 \), we can set up the equation:
\[ \frac{\sqrt{3}}{4} a^2 = 81 \sqrt{3} \]

### Step 2: Eliminate \( \sqrt{3} \) from both sides.
To simplify the equation, we can divide both sides by \( \sqrt{3} \):
\[ \frac{1}{4} a^2 = 81 \]

### Step 3: Solve for \( a^2 \).
Multiply both sides by 4 to isolate \( a^2 \):
\[ a^2 = 81 \times 4 \]
\[ a^2 = 324 \]

### Step 4: Find \( a \) by taking the square root.
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{324} = 18 \, \text{cm} \]

### Step 5: Use the height formula for an equilateral triangle.
The height \( h \) of an equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]
Substituting \( a = 18 \, \text{cm} \):
\[ h = \frac{\sqrt{3}}{2} \times 18 \]
\[ h = 9 \sqrt{3} \, \text{cm} \]

### Conclusion
The height of the equilateral triangle is:
\[ h = 9 \sqrt{3} \, \text{cm} \]

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