Find the volume of the right circular cone with (i) radius 6 cm, height 7 cm (ii) radius 3.5 cm, height 12 cm

Answer

To find the volume of the right circular cone for the given dimensions, we will use the formula for the volume of a cone:

\[ \text{Volume of cone} = \frac{1}{3} \pi r^2 h \]

where:
- \( r \) is the radius of the cone,
- \( h \) is the height of the cone,
- \( \pi \) is approximately \( \frac{22}{7} \).

### Part (i): Radius = 6 cm, Height = 7 cm

1. **Identify the values**:
- Radius \( r = 6 \) cm
- Height \( h = 7 \) cm

2. **Substitute the values into the formula**:
\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times (6^2) \times 7 \]

3. **Calculate \( r^2 \)**:
\[ 6^2 = 36 \]

4. **Substitute \( r^2 \) back into the formula**:
\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times 36 \times 7 \]

5. **Simplify the equation**:
- The \( 7 \) in the numerator and denominator cancels out:
\[ \text{Volume} = \frac{1}{3} \times 22 \times 36 \]

6. **Calculate \( 22 \times 36 \)**:
\[ 22 \times 36 = 792 \]

7. **Now divide by 3**:
\[ \text{Volume} = \frac{792}{3} = 264 \text{ cm}^3 \]

### Part (ii): Radius = 3.5 cm, Height = 12 cm

1. **Identify the values**:
- Radius \( r = 3.5 \) cm
- Height \( h = 12 \) cm

2. **Substitute the values into the formula**:
\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times (3.5^2) \times 12 \]

3. **Calculate \( r^2 \)**:
\[ 3.5^2 = 12.25 \]

4. **Substitute \( r^2 \) back into the formula**:
\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times 12 \]

5. **Simplify the equation**:
- First, calculate \( 12.25 \times 12 \):
\[ 12.25 \times 12 = 147 \]
- Now substitute back:
\[ \text{Volume} = \frac{1}{3} \times \frac{22}{7} \times 147 \]

6. **Calculate \( \frac{147}{3} \)**:
\[ \frac{147}{3} = 49 \]

7. **Now multiply by \( \frac{22}{7} \)**:
\[ \text{Volume} = \frac{22 \times 49}{7} \]

8. **Calculate \( 22 \times 49 \)**:
\[ 22 \times 49 = 1078 \]

9. **Now divide by 7**:
\[ \text{Volume} = \frac{1078}{7} = 154 \text{ cm}^3 \]

### Final Answers:
- Volume of the first cone: \( 264 \text{ cm}^3 \)
- Volume of the second cone: \( 154 \text{ cm}^3 \)