Find the ratio of in radius and exradius in an equilateral triangle

Answer

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

### Step 1: Define the side length of the equilateral triangle
Let the side length of the equilateral triangle be \( a \).

### Step 2: Calculate the inradius (r)
The formula for the inradius \( r \) of an equilateral triangle is given by:
\[ r = \frac{a}{2\sqrt{3}} \]

### Step 3: Calculate the exradius (R)
The exradius (or circumradius) \( R \) of an equilateral triangle is given by:
\[ R = \frac{a}{\sqrt{3}} \]

### Step 4: Find the ratio of inradius to exradius
Now, we need to find the ratio of the inradius \( r \) to the exradius \( R \):
\[ \text{Ratio} = \frac{r}{R} = \frac{\frac{a}{2\sqrt{3}}}{\frac{a}{\sqrt{3}}} \]

### Step 5: Simplify the ratio
When we simplify the ratio, we can cancel \( a \) from the numerator and denominator:
\[ \text{Ratio} = \frac{1}{2} \]

### Final Answer
Thus, the ratio of the inradius to the exradius in an equilateral triangle is:
\[ \frac{1}{2} \]

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