If the incentre of an equilateral triangle lies inside the triangle and its radius Is 3 cm, then the side of the equilateral triangle is

Options

A. `9sqrt(3)` cm
B. `6sqrt(3)` cm
C. `3sqrt(3)` cm
D. `6` cm

Answer

To find the side length of an equilateral triangle given its inradius, we can use the relationship between the inradius (r) and the side length (a) of an equilateral triangle. The formula is:

\[ r = \frac{a \sqrt{3}}{6} \]

Where:
- \( r \) is the inradius
- \( a \) is the side length of the equilateral triangle

Given that the inradius \( r = 3 \) cm, we can substitute this value into the formula and solve for \( a \).

### Step-by-Step Solution:

1. **Write the formula for the inradius of an equilateral triangle:**
\[ r = \frac{a \sqrt{3}}{6} \]

2. **Substitute the given inradius into the formula:**
\[ 3 = \frac{a \sqrt{3}}{6} \]

3. **Multiply both sides by 6 to eliminate the fraction:**
\[ 18 = a \sqrt{3} \]

4. **Now, divide both sides by \( \sqrt{3} \) to solve for \( a \):**
\[ a = \frac{18}{\sqrt{3}} \]

5. **Rationalize the denominator:**
\[ a = \frac{18 \sqrt{3}}{3} = 6 \sqrt{3} \]

6. **Thus, the side length of the equilateral triangle is:**
\[ a = 6\sqrt{3} \text{ cm} \]

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