The area of the circumcircle of an equilateral triangle is `3pi` sq. cm. The perimeter of the triangle is

Options

A. `3sqrt(3)cm`
B. 9 cm
C. 18 cm
D. 3 cm

Answer

To find the perimeter of the equilateral triangle given that the area of its circumcircle is \(3\pi\) square centimeters, we can follow these steps:

### Step 1: Understand the relationship between the circumradius and the side of the triangle
For an equilateral triangle, the circumradius \(R\) is related to the side length \(s\) by the formula:
\[ R = \frac{s}{\sqrt{3}} \]

### Step 2: Write the formula for the area of the circumcircle
The area \(A\) of the circumcircle can be expressed as:
\[ A = \pi R^2 \]

### Step 3: Substitute the expression for \(R\) into the area formula
Substituting \(R = \frac{s}{\sqrt{3}}\) into the area formula gives:
\[ A = \pi \left(\frac{s}{\sqrt{3}}\right)^2 = \pi \frac{s^2}{3} \]

### Step 4: Set the area equal to the given area
We know from the problem that the area of the circumcircle is \(3\pi\):
\[ \pi \frac{s^2}{3} = 3\pi \]

### Step 5: Simplify the equation
Dividing both sides by \(\pi\) (assuming \(\pi \neq 0\)):
\[ \frac{s^2}{3} = 3 \]

### Step 6: Solve for \(s^2\)
Multiplying both sides by 3 gives:
\[ s^2 = 9 \]

### Step 7: Find the side length \(s\)
Taking the square root of both sides:
\[ s = \sqrt{9} = 3 \text{ cm} \]

### Step 8: Calculate the perimeter of the triangle
The perimeter \(P\) of an equilateral triangle is given by:
\[ P = 3s \]
Substituting \(s = 3\):
\[ P = 3 \times 3 = 9 \text{ cm} \]

### Final Answer
The perimeter of the triangle is \(9\) cm.

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