If the area of circumcirle of an equilateral triangle is `(25pi)/2 cm^(2)`, then find the radius of the incircle to the triangle.

Answer

To find the radius of the incircle of an equilateral triangle given the area of its circumcircle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Area of the Circumcircle**:
The area of the circumcircle is given as \( \frac{25\pi}{2} \, \text{cm}^2 \).

2. **Use the Area Formula for a Circle**:
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.

3. **Set Up the Equation**:
From the area of the circumcircle, we can set up the equation:
\[ \pi R^2 = \frac{25\pi}{2} \]
where \( R \) is the radius of the circumcircle.

4. **Solve for \( R^2 \)**:
Dividing both sides by \( \pi \):
\[ R^2 = \frac{25}{2} \]

5. **Calculate \( R \)**:
Taking the square root of both sides:
\[ R = \sqrt{\frac{25}{2}} = \frac{5}{\sqrt{2}} \, \text{cm} \]

6. **Relate the Circumradius to the Side Length**:
For an equilateral triangle, the circumradius \( R \) is related to the side length \( a \) by the formula:
\[ R = \frac{a}{\sqrt{3}} \]
Thus, we can set:
\[ \frac{5}{\sqrt{2}} = \frac{a}{\sqrt{3}} \]

7. **Solve for \( a \)**:
Rearranging gives:
\[ a = \frac{5\sqrt{3}}{\sqrt{2}} = \frac{5\sqrt{6}}{2} \, \text{cm} \]

8. **Find the Inradius**:
The radius of the incircle \( r \) of an equilateral triangle is given by:
\[ r = \frac{a}{2\sqrt{3}} \]
Substituting in the value of \( a \):
\[ r = \frac{\frac{5\sqrt{6}}{2}}{2\sqrt{3}} = \frac{5\sqrt{6}}{4\sqrt{3}} = \frac{5\sqrt{2}}{4} \, \text{cm} \]

### Final Answer:
The radius of the incircle of the triangle is \( \frac{5\sqrt{2}}{4} \, \text{cm} \).

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