If the radius of the circumcircle of an equilateral triangle is 6 cm , then find the radius of its incircle.

Answer

To find the radius of the incircle of an equilateral triangle when the radius of the circumcircle is given, we can use the relationship between the circumradius (R) and the inradius (r) of an equilateral triangle.

### Step-by-Step Solution:

1. **Understand the Relationship**:
For an equilateral triangle, the relationship between the circumradius (R) and the inradius (r) is given by the formula:
\[ R = \frac{2r}{\sqrt{3}} \]
Rearranging this formula to express r in terms of R gives:
\[ r = \frac{R \cdot \sqrt{3}}{2} \]

2. **Substitute the Given Value**:
We know that the radius of the circumcircle (R) is 6 cm. Substitute this value into the rearranged formula:
\[ r = \frac{6 \cdot \sqrt{3}}{2} \]

3. **Simplify the Expression**:
Now simplify the expression:
\[ r = 3 \cdot \sqrt{3} \]

4. **Calculate the Numerical Value (if needed)**:
If you need a numerical approximation, you can calculate:
\[ \sqrt{3} \approx 1.732 \]
Therefore,
\[ r \approx 3 \cdot 1.732 \approx 5.196 \text{ cm} \]

5. **Final Answer**:
The radius of the incircle of the equilateral triangle is:
\[ r = 3\sqrt{3} \text{ cm} \quad \text{or approximately } 5.196 \text{ cm} \]