The height and the slant height of a cone are 21 cm and 28 cm respectively. Find the volume of the cone.

Answer

To find the volume of the cone given the height and slant height, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Height (h) of the cone = 21 cm
- Slant height (l) of the cone = 28 cm

2. **Use the Pythagorean Theorem to Find the Radius (r):**
The relationship between the radius, height, and slant height of a cone can be expressed using the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \]
Substituting the known values:
\[ 28^2 = r^2 + 21^2 \]
This simplifies to:
\[ 784 = r^2 + 441 \]
Rearranging gives:
\[ r^2 = 784 - 441 = 343 \]

3. **Calculate the Radius:**
Taking the square root of both sides:
\[ r = \sqrt{343} = 7\sqrt{7} \text{ cm} \]

4. **Use the Volume Formula for a Cone:**
The formula for the volume (V) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V = \frac{1}{3} \times \frac{22}{7} \times (7\sqrt{7})^2 \times 21 \]

5. **Calculate \( r^2 \):**
\[ (7\sqrt{7})^2 = 49 \times 7 = 343 \]

6. **Substituting Back into the Volume Formula:**
\[ V = \frac{1}{3} \times \frac{22}{7} \times 343 \times 21 \]

7. **Simplifying the Volume Calculation:**
- First, calculate \( \frac{343 \times 21}{7} \):
\[ \frac{343 \times 21}{7} = 343 \times 3 = 1029 \]
- Now, multiply by \( \frac{22}{3} \):
\[ V = \frac{22 \times 1029}{3} = 22 \times 343 = 7546 \text{ cm}^3 \]

### Final Answer:
The volume of the cone is \( 7546 \text{ cm}^3 \).