The height of an equilateral triangle is `6 cm.` Then the area of the triangle is

Options

A. `15sqrt(3)cm^(2)`
B. `32sqrt(3)cm^(2)`
C. `12sqrt(3)cm^(2)`
D. `17sqrt(3)cm^(2)`

Answer

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

### Step 2: Substitute the given height into the formula
Given that the height \( h = 6 \) cm, we can set up the equation:
\[ 6 = \frac{\sqrt{3}}{2} \cdot a \]

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

### Step 4: Calculate the area of the equilateral triangle
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} \cdot a^2 \]
Substituting \( a = \frac{12}{\sqrt{3}} \):
\[ A = \frac{\sqrt{3}}{4} \cdot \left(\frac{12}{\sqrt{3}}\right)^2 \]

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

### Final Answer
Thus, the area of the equilateral triangle is:
\[ \text{Area} = 12\sqrt{3} \text{ cm}^2 \]

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