The circum-radius of an equilateral triangle is 8 cm. The in-radius of the triangle is

Options

A. 3.25 cm
B. 3.50 cm
C. 4 cm
D. 4.25 cm

Answer

To find the in-radius of an equilateral triangle given its circum-radius, we can follow these steps:

### Step 1: Understand the relationship between circum-radius (R) and in-radius (r)
For an equilateral triangle, the relationship between the circum-radius (R) and in-radius (r) is given by the formula:
\[ r = \frac{R \sqrt{3}}{3} \]

### Step 2: Substitute the given circum-radius value
We are given that the circum-radius (R) is 8 cm. Substitute this value into the formula:
\[ r = \frac{8 \sqrt{3}}{3} \]

### Step 3: Calculate the in-radius
Now, we can calculate the in-radius (r):
1. First, calculate \( 8 \sqrt{3} \):
\[ 8 \sqrt{3} \approx 8 \times 1.732 \approx 13.856 \]

2. Now divide this by 3:
\[ r \approx \frac{13.856}{3} \approx 4.6187 \]

### Step 4: Round to a suitable precision
Since we need the in-radius in cm, we can round it to two decimal places:
\[ r \approx 4.62 \text{ cm} \]

### Final Answer
The in-radius of the triangle is approximately \( 4.62 \text{ cm} \).

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