The base of a pyramid is a rectangle whose length and breadth are 16 cm and 12 cm respectively. If the length of all the lateral edges passing through the vertex of the right rectangular pyramid is 26 cm, then find the volume of the pyramid in cubic centimeter?

Options

A. 1536
B. 1024
C. 718
D. 2072

Answer

To find the volume of the pyramid, we can follow these steps:

### Step 1: Identify the dimensions of the base
The base of the pyramid is a rectangle with:
- Length (L) = 16 cm
- Breadth (B) = 12 cm

### Step 2: Calculate the diagonal of the base
The diagonal (D) of the rectangle can be calculated using the Pythagorean theorem:
\[ D = \sqrt{L^2 + B^2} \]
Substituting the values:
\[ D = \sqrt{16^2 + 12^2} \]
\[ D = \sqrt{256 + 144} \]
\[ D = \sqrt{400} \]
\[ D = 20 \text{ cm} \]

### Step 3: Relate the lateral edge to the height of the pyramid
Let the height of the pyramid be \( H \). The lateral edge (E) of the pyramid is given as 26 cm. The relationship between the lateral edge, height, and half of the diagonal of the base can be expressed as:
\[ E^2 = H^2 + \left(\frac{D}{2}\right)^2 \]
Substituting the known values:
\[ 26^2 = H^2 + \left(\frac{20}{2}\right)^2 \]
\[ 676 = H^2 + 10^2 \]
\[ 676 = H^2 + 100 \]

### Step 4: Solve for the height (H)
Rearranging the equation to solve for \( H^2 \):
\[ H^2 = 676 - 100 \]
\[ H^2 = 576 \]
Taking the square root:
\[ H = \sqrt{576} \]
\[ H = 24 \text{ cm} \]

### Step 5: Calculate the volume of the pyramid
The volume (V) of the pyramid can be calculated using the formula:
\[ V = \frac{1}{3} \times \text{Area of base} \times \text{Height} \]
The area of the base (A) is:
\[ A = L \times B = 16 \times 12 = 192 \text{ cm}^2 \]
Now substituting the area and height into the volume formula:
\[ V = \frac{1}{3} \times 192 \times 24 \]
Calculating:
\[ V = \frac{1}{3} \times 4608 \]
\[ V = 1536 \text{ cm}^3 \]

### Final Answer
The volume of the pyramid is **1536 cm³**.

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