The volume of a right circular cone of height `12 cm` and base radius `6 cm,` is

Options

A. `(12pi) cm^(3)`
B. `(36pi) cm^(3)`
C. `(72 pi) cm^(3)`
D. `(144 pi)cm^(3)`

Answer

To find the volume of a right circular cone with a height of 12 cm and a base radius of 6 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formula for the volume of a cone**:
The volume \( V \) of a right circular cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height of the cone.

2. **Substitute the values into the formula**:
Here, the radius \( r = 6 \) cm and the height \( h = 12 \) cm. Plugging these values into the formula gives:
\[ V = \frac{1}{3} \pi (6)^2 (12) \]

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

4. **Substitute \( r^2 \) back into the volume formula**:
Now substitute \( r^2 \) back into the equation:
\[ V = \frac{1}{3} \pi (36)(12) \]

5. **Multiply \( 36 \) and \( 12 \)**:
Calculate the product:
\[ 36 \times 12 = 432 \]

6. **Now substitute this product back into the volume formula**:
\[ V = \frac{1}{3} \pi (432) \]

7. **Divide \( 432 \) by \( 3 \)**:
Now divide \( 432 \) by \( 3 \):
\[ \frac{432}{3} = 144 \]

8. **Final volume expression**:
Therefore, the volume of the cone is:
\[ V = 144 \pi \text{ cm}^3 \]

### Final Answer:
The volume of the right circular cone is \( 144 \pi \text{ cm}^3 \).

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