The inradius of an equilateral triangle is `sqrt(3)cm`, then the perimeter of that triangle is

Options

A. 18 cm
B. 15 cm
C. 12 cm
D. 6 cm

Answer

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

### Step 1: Understand the relationship between the inradius and the side length of an equilateral triangle.
The formula for the inradius \( r \) of an equilateral triangle with side length \( a \) is given by:
\[ r = \frac{a \sqrt{3}}{6} \]

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

### Step 3: Solve for the side length \( a \).
To eliminate \( \sqrt{3} \) from both sides, we can multiply both sides by 6:
\[ 6\sqrt{3} = a \sqrt{3} \]
Now, divide both sides by \( \sqrt{3} \):
\[ a = 6 \]

### Step 4: Calculate the perimeter of the triangle.
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting the value of \( a \):
\[ P = 3 \times 6 = 18 \text{ cm} \]

### Final Answer:
The perimeter of the equilateral triangle is \( 18 \) cm.

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