Question

The base of a right pyramid is a square and the length of the side of the square is 32 cm, and the height of the pyramid is 12 cm. Then what is the total surface area of the square pyramid?

• A. 2114 sq. cm
• B. 2304 sq. cm
• C. 2204 sq. cm
• D. 2314 sq. cm

Hint:

To calculate the total surface area of a square pyramid, first find the slant height using the Pythagorean theorem. Then, calculate both the base area and the lateral surface area and add them together.

Answer 1

The Correct Option is B

Step 1: Calculate the base area of the square. The side of the square is given as 32 cm. Therefore, the base area \( A \) is: \[ A = \text{side}^2 = 32 \times 32 = 1024 \, \text{sq. cm} \]

Step 2: Find the slant height of the pyramid. The slant height \( l \) is the hypotenuse of a right triangle formed by half the side of the base (16 cm) and the height of the pyramid (12 cm). Using the Pythagorean theorem: \[ l = \sqrt{(16)^2 + (12)^2} = \sqrt{256 + 144} = \sqrt{400} = 20 \, \text{cm} \]

Step 3: Calculate the lateral surface area. The lateral surface area (C.S.A) of the pyramid is given by: \[ \text{C.S.A} = \frac{1}{2} \times \text{Base Perimeter} \times \text{Slant Height} \] The perimeter of the base is: \[ \text{Base Perimeter} = 4 \times \text{side} = 4 \times 32 = 128 \, \text{cm} \] Thus, the lateral surface area is: \[ \text{C.S.A} = \frac{1}{2} \times 128 \times 20 = 1280 \, \text{sq. cm} \]

Step 4: Calculate the total surface area. The total surface area (T.S.A) of the pyramid is the sum of the lateral surface area and the base area: \[ \text{T.S.A} = \text{C.S.A} + \text{Base Area} = 1280 + 1024 = 2304 \, \text{sq. cm} \] Final Answer: The correct answer is (b) 2304 sq. cm.