Each edge of an equilateral triangle is 'a' cm. A cone is formed by joining any two sides of the triangle.
What is the radius and slant height of the cone ?

Options

A. `a,(a)/(2pi) `
B. `(a)/(pi),(a)/(2) `
C. `(a)/(2pi),a`
D. `2a,(a)/(pi)`

Answer

To find the radius and slant height of the cone formed by joining two sides of an equilateral triangle with each edge measuring 'a' cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure**:
- We have an equilateral triangle ABC where each side is 'a' cm.
- When we join two sides, say AC and BC, we form a cone with the base at point B and the apex at point A.

2. **Identify the Slant Height**:
- The slant height of the cone is equal to the length of the sides of the triangle.
- Therefore, the slant height (l) of the cone is:
\[ l = a \text{ cm} \]

3. **Calculate the Perimeter of the Base Circle**:
- The perimeter (circumference) of the base circle of the cone is equal to the length of the side AB of the triangle, which is also 'a' cm.
- Hence, the circumference (C) of the base circle is:
\[ C = a \text{ cm} \]

4. **Relate the Circumference to the Radius**:
- The circumference of a circle is given by the formula:
\[ C = 2\pi r \]
- Setting the two expressions for circumference equal gives:
\[ 2\pi r = a \]

5. **Solve for the Radius**:
- To find the radius (r) of the base of the cone, we rearrange the equation:
\[ r = \frac{a}{2\pi} \]

6. **Final Results**:
- Thus, the radius of the cone is:
\[ r = \frac{a}{2\pi} \text{ cm} \]
- The slant height of the cone is:
\[ l = a \text{ cm} \]

### Summary of Results:
- **Radius of the cone**: \( r = \frac{a}{2\pi} \) cm
- **Slant height of the cone**: \( l = a \) cm