If the radius of a circumscribed cirlce in an equilateral triangle is 14 cm. Then what will be the radius (in cm) of the inscribed circle ?

Options

A. 7
B. 14
C. 28
D. 35

Answer

To find the radius of the inscribed circle (inradius) of an equilateral triangle when the radius of the circumscribed circle (circumradius) is given, we can use the relationship between the two radii.

### Step-by-Step Solution:

1. **Understand the Relationship**: In an equilateral triangle, the circumradius \( R \) and the inradius \( r \) are related by the formula:
\[ R = \frac{2r}{\sqrt{3}} \]
This means that the circumradius is proportional to the inradius.

2. **Given Information**: We are given that the circumradius \( R \) is 14 cm.

3. **Set Up the Equation**: We can rearrange the formula to find \( r \):
\[ r = \frac{R \cdot \sqrt{3}}{2} \]

4. **Substitute the Value of \( R \)**: Now, substitute \( R = 14 \) cm into the equation:
\[ r = \frac{14 \cdot \sqrt{3}}{2} \]

5. **Simplify the Expression**:
\[ r = 7 \cdot \sqrt{3} \]

6. **Calculate the Value of \( r \)**: To find the numerical value of \( r \):
\[ r \approx 7 \cdot 1.732 \approx 12.124 \text{ cm} \]

7. **Final Result**: The radius of the inscribed circle is approximately \( 12.12 \) cm.