The area of an equilateral triangle is `16sqrt(3)` sq cm. What is its perimeter?

Options

A. 16 cm
B. 12 cm
C. 24 cm
D. 32 cm

Answer

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

### Step 1: Write 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:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

### Step 2: Set the area equal to the given value.
We know the area of the triangle is \( 16\sqrt{3} \) sq cm. Therefore, we can set up the equation:
\[ \frac{\sqrt{3}}{4} a^2 = 16\sqrt{3} \]

### Step 3: Eliminate the square root from both sides.
To simplify the equation, we can multiply both sides by \( 4 \):
\[ \sqrt{3} a^2 = 64\sqrt{3} \]

### Step 4: Divide both sides by \( \sqrt{3} \).
This allows us to cancel out \( \sqrt{3} \):
\[ a^2 = 64 \]

### Step 5: Solve for \( a \).
Taking the square root of both sides gives us:
\[ a = \sqrt{64} = 8 \text{ cm} \]

### Step 6: Calculate the perimeter of the triangle.
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting the value of \( a \):
\[ P = 3 \times 8 = 24 \text{ cm} \]

### Final Answer:
The perimeter of the equilateral triangle is \( 24 \) cm.

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