The ratio of circumradius and In radius of an equilateral triangle is

Options

A. `1:2`
B. `3:1 `
C. `2:1 `
D. `1:3`

Answer

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

### Step 1: Understand the Definitions
- **Circumradius (R)**: The radius of the circumcircle, which is the circle that passes through all three vertices of the triangle.
- **Inradius (r)**: The radius of the incircle, which is the circle that touches all three sides of the triangle.

### Step 2: Properties of an Equilateral Triangle
In an equilateral triangle, the circumradius (R) and inradius (r) can be expressed in terms of the side length (a) of the triangle:
- The circumradius \( R \) is given by the formula:
\[ R = \frac{a}{\sqrt{3}} \]
- The inradius \( r \) is given by the formula:
\[ r = \frac{a}{2\sqrt{3}} \]

### Step 3: Calculate the Ratio \( \frac{R}{r} \)
Now, we can find the ratio of the circumradius to the inradius:
\[ \frac{R}{r} = \frac{\frac{a}{\sqrt{3}}}{\frac{a}{2\sqrt{3}}} \]

### Step 4: Simplify the Ratio
When we simplify the ratio:
\[ \frac{R}{r} = \frac{a}{\sqrt{3}} \times \frac{2\sqrt{3}}{a} = \frac{2\sqrt{3}}{\sqrt{3}} = 2 \]

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

### Final Answer
The ratio of circumradius to inradius of an equilateral triangle is \( 2:1 \).

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