If the base of a right pyramid is triangle of sides 5 cm, 12 cm, 13 cm and its volume is `330 cm^(3)`, then its height (in cm) will be

Options

A. 33
B. 32
C. 11
D. 22

Answer

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

### Step 1: Identify the base area of the triangular base
Given the sides of the triangle are 5 cm, 12 cm, and 13 cm, we can confirm that this is a right-angled triangle (5² + 12² = 13²).

The area \( A \) of a right triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take the base as 12 cm and the height as 5 cm (or vice versa). Thus, the area of the triangle is:
\[ A = \frac{1}{2} \times 12 \times 5 = 30 \, \text{cm}^2 \]

### Step 2: Use the volume formula of the pyramid
The volume \( V \) of a pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
We know the volume \( V \) is 330 cm³ and the base area \( A \) is 30 cm². Let the height of the pyramid be \( h \).

### Step 3: Set up the equation
Substituting the known values into the volume formula gives us:
\[ 330 = \frac{1}{3} \times 30 \times h \]

### Step 4: Solve for height \( h \)
To isolate \( h \), we can first multiply both sides by 3:
\[ 990 = 30 \times h \]
Now, divide both sides by 30:
\[ h = \frac{990}{30} = 33 \, \text{cm} \]

### Conclusion
The height of the pyramid is \( 33 \, \text{cm} \).

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