If the length of the side of an equilateral triangle is 12 cm, then what is its in-radius?

Answer

To find the in-radius (r) of an equilateral triangle with a given side length (a), we can use the formula:

\[ r = \frac{a}{2\sqrt{3}} \]

### Step-by-Step Solution:

1. **Identify the side length of the equilateral triangle:**
Given that the length of the side (a) is 12 cm.

2. **Substitute the side length into the formula for in-radius:**
\[ r = \frac{12}{2\sqrt{3}} \]

3. **Simplify the expression:**
First, calculate the denominator:
\[ 2\sqrt{3} \approx 2 \times 1.732 \approx 3.464 \]
Now, substitute this back into the equation:
\[ r = \frac{12}{3.464} \]

4. **Perform the division:**
\[ r \approx 3.464 \text{ cm} \]

5. **Final result:**
The in-radius of the equilateral triangle is approximately 3.464 cm.