The height of an equilateral triangle is `sqrt6` cm. Its area is

Options

A. `3sqrt3cm`
B. `2sqrt3cm^(2)`
C. `2sqrt2`
D. `6sqrt2cm^(2)`

Answer

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

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

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

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

### Step 4: Simplify the expression for \( a \).
We can simplify \( a \):
\[ a = 2 \cdot \frac{\sqrt{6}}{\sqrt{3}} = 2 \cdot \sqrt{\frac{6}{3}} = 2 \cdot \sqrt{2} = 2\sqrt{2} \, \text{cm} \]

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

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

### Step 7: Calculate \( (2\sqrt{2})^2 \).
Calculating this gives:
\[ (2\sqrt{2})^2 = 4 \cdot 2 = 8 \]

### Step 8: Substitute back into the area formula.
Now substitute back:
\[ A = \frac{\sqrt{3}}{4} \cdot 8 \]

### Step 9: Simplify the area.
This simplifies to:
\[ A = 2\sqrt{3} \, \text{cm}^2 \]

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

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