If in-radius of an equilateral triangle is 3 cm. Calculate the perimeter (in cm) of the equilateral triangle.

Options

A. `6 sqrt(3)`
B. `3 sqrt(3)`
C. `12 sqrt(3)`
D. `18 sqrt(3)`

Answer

To find the perimeter of an equilateral triangle given its in-radius (r), we can use the relationship between the in-radius and the side length of the triangle. Here are the step-by-step calculations:

### Step 1: Understand the relationship between in-radius and side length
For an equilateral triangle, the in-radius (r) can be expressed in terms of the side length (a) as follows:
\[ r = \frac{a \sqrt{3}}{6} \]

### Step 2: Substitute the given in-radius
We are given that the in-radius \( r = 3 \) cm. We can substitute this value into the equation:
\[ 3 = \frac{a \sqrt{3}}{6} \]

### Step 3: Solve for the side length (a)
To find the side length \( a \), we can rearrange the equation:
\[ a \sqrt{3} = 3 \times 6 \]
\[ a \sqrt{3} = 18 \]
Now, divide both sides by \( \sqrt{3} \):
\[ a = \frac{18}{\sqrt{3}} \]

### Step 4: Simplify the side length (a)
To simplify \( a \), we can multiply the numerator and denominator by \( \sqrt{3} \):
\[ a = \frac{18 \sqrt{3}}{3} = 6\sqrt{3} \]

### Step 5: Calculate the perimeter
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting the value of \( a \):
\[ P = 3 \times 6\sqrt{3} = 18\sqrt{3} \]

### Final Answer
The perimeter of the equilateral triangle is \( 18\sqrt{3} \) cm.

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