A triangle with three equal sides has its area equal to `3sqrt(3)` sq cm. Find its perimeter.

Options

A. `6 sqrt(3)`
B. `2 sqrt(3)`
C. `5 sqrt(3)`
D. `7 sqrt(3)`

Answer

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

### Step 1: Understand 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 \( 3\sqrt{3} \) sq cm. Therefore, we can set up the equation:
\[ \frac{\sqrt{3}}{4} a^2 = 3\sqrt{3} \]

### Step 3: Solve for \( a^2 \).
To eliminate \(\sqrt{3}\) from both sides, we can multiply both sides by \( 4 \):
\[ \sqrt{3} a^2 = 12\sqrt{3} \]
Now, divide both sides by \(\sqrt{3}\):
\[ a^2 = 12 \]

### Step 4: Find \( a \).
To find \( a \), take the square root of both sides:
\[ a = \sqrt{12} = 2\sqrt{3} \]

### Step 5: Calculate the perimeter.
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting the value of \( a \):
\[ P = 3 \times 2\sqrt{3} = 6\sqrt{3} \]

### Final Answer:
The perimeter of the triangle is \( 6\sqrt{3} \) cm.

---