The area of an equilateral triangle is `144 sqrt(3) cm^(2)`, find its perimeter.

Answer

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

### Step 1: Use the formula for the area of an equilateral triangle.
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
where \( a \) is the length of a side of the triangle.

### Step 2: Substitute the given area into the formula.
We know the area of the triangle is \( 144 \sqrt{3} \, \text{cm}^2 \). We can set up the equation:
\[ 144 \sqrt{3} = \frac{\sqrt{3}}{4} a^2 \]

### Step 3: Simplify the equation.
To simplify, we can divide both sides by \( \sqrt{3} \):
\[ 144 = \frac{1}{4} a^2 \]

### Step 4: Multiply both sides by 4 to isolate \( a^2 \).
\[ 144 \times 4 = a^2 \]
\[ 576 = a^2 \]

### Step 5: Take the square root of both sides to find \( a \).
\[ a = \sqrt{576} \]
\[ a = 24 \, \text{cm} \]

### Step 6: Calculate the perimeter of the triangle.
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting the value of \( a \):
\[ P = 3 \times 24 \]
\[ P = 72 \, \text{cm} \]

### Final Answer:
The perimeter of the equilateral triangle is \( 72 \, \text{cm} \).

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