The ratio of inradius and circum radius of an equilateral triangle is :

Options

A. `1:2`
B. `2:1`
C. `1: sqrt(2)`
D. `sqrt(2):1`

Answer

To find the ratio of the inradius (r) and circumradius (R) of an equilateral triangle, we can follow these steps:

### Step 1: Understand the definitions
- **Inradius (r)**: The radius of the circle inscribed in the triangle.
- **Circumradius (R)**: The radius of the circle circumscribed around the triangle.

### Step 2: Use the formulas for inradius and circumradius
For an equilateral triangle with side length \( A \):
- The formula for the inradius \( r \) is:
\[ r = \frac{A}{2\sqrt{3}} \]
- The formula for the circumradius \( R \) is:
\[ R = \frac{A}{\sqrt{3}} \]

### Step 3: Set up the ratio of inradius to circumradius
Now, we need to find the ratio \( \frac{r}{R} \):
\[ \frac{r}{R} = \frac{\frac{A}{2\sqrt{3}}}{\frac{A}{\sqrt{3}}} \]

### Step 4: Simplify the ratio
When we simplify the ratio, the \( A \) cancels out:
\[ \frac{r}{R} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the inradius to the circumradius of an equilateral triangle is:
\[ \frac{r}{R} = \frac{1}{2} \]