The height of an equilateral triangle is 6 cm. Its area is

Options

A. `12 sqrt(3) cm^(2)`
B. `6 sqrt(3) cm^(2)`
C. `12 sqrt(2) cm^(2)`
D. 18 `cm^(2)`

Explanation

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

### Step 1: Understand the relationship between the height and the side of the equilateral triangle.
For an equilateral triangle with side length \( a \), the height \( h \) can be expressed as:
\[ h = \frac{\sqrt{3}}{2} a \]

### Step 2: Substitute the given height into the equation.
We know the height \( h = 6 \) cm. Therefore, we can set up the equation:
\[ 6 = \frac{\sqrt{3}}{2} a \]

### Step 3: Solve for the side length \( a \).
To find \( a \), we can rearrange the equation:
\[ a = \frac{6 \times 2}{\sqrt{3}} = \frac{12}{\sqrt{3}} \text{ cm} \]

### Step 4: Calculate the area of the equilateral triangle.
The area \( A \) of an equilateral triangle is given by the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting the value of \( a \):
\[ A = \frac{\sqrt{3}}{4} \left(\frac{12}{\sqrt{3}}\right)^2 \]

### Step 5: Simplify the expression.
Calculating \( a^2 \):
\[ \left(\frac{12}{\sqrt{3}}\right)^2 = \frac{144}{3} = 48 \]
Now substituting back into the area formula:
\[ A = \frac{\sqrt{3}}{4} \times 48 = 12\sqrt{3} \text{ cm}^2 \]

### Step 6: Calculate the numerical value of the area.
Using \( \sqrt{3} \approx 1.732 \):
\[ A \approx 12 \times 1.732 \approx 20.784 \text{ cm}^2 \]

### Final Answer:
The area of the equilateral triangle is approximately \( 20.78 \text{ cm}^2 \).

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