What is the volume (in `cm^(3)`) of a right pyramid of height 12 cm and having a square base whose diagonal is `6sqrt(2)` cm ?

Options

A. 864
B. 432
C. 144
D. 288

Answer

To find the volume of a right pyramid with a square base, we can follow these steps:

### Step 1: Find the side length of the square base
The diagonal of the square base is given as \(6\sqrt{2}\) cm. We can use the relationship between the side length \(a\) of a square and its diagonal \(d\):
\[ d = a\sqrt{2} \]
Given \(d = 6\sqrt{2}\), we can set up the equation:
\[ 6\sqrt{2} = a\sqrt{2} \]
To find \(a\), divide both sides by \(\sqrt{2}\):
\[ a = 6 \text{ cm} \]

### Step 2: Calculate the area of the base
The area \(A\) of a square is given by the formula:
\[ A = a^2 \]
Substituting the value of \(a\):
\[ A = 6^2 = 36 \text{ cm}^2 \]

### Step 3: Use the volume formula for the pyramid
The volume \(V\) of a pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Area of base} \times \text{Height} \]
Substituting the area of the base and the height (which is given as 12 cm):
\[ V = \frac{1}{3} \times 36 \times 12 \]

### Step 4: Calculate the volume
Now, calculate the volume:
\[ V = \frac{1}{3} \times 36 \times 12 = \frac{432}{3} = 144 \text{ cm}^3 \]

### Final Answer
The volume of the pyramid is \(144 \text{ cm}^3\).

---