Question:
A square pyramid with a side of 10 cm has a volume of 400 cm\(^3\). Find the total surface area of the pyramid.
A square pyramid with a side of 10 cm has a volume of 400 cm\(^3\). Find the total surface area of the pyramid.
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For pyramid problems, always start by using the formula for volume to find the height, then use the Pythagorean theorem to find the slant height. The total surface area combines both the base area and lateral surface area.
Updated On: Mar 10, 2025
- 240 cm\(^2\)
- 300 cm\(^2\)
- 360 cm\(^2\)
- 420 cm\(^2\)
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The Correct Option is C
Solution and Explanation
The volume of the pyramid is given as 400 cm\(^3\). The formula for the volume of a square pyramid is:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
We are given that the base side of the pyramid is 10 cm. Therefore, the base area is:
\[
\text{Base Area} = \text{side}^2 = 10 \times 10 = 100 \, \text{cm}^2
\]
Substitute this value into the volume formula:
\[
400 = \frac{1}{3} \times 100 \times \text{Height}
\]
Solving for the height:
\[
\text{Height} = \frac{400 \times 3}{100} = 12 \, \text{cm}
\]
Now that we have the height, we can calculate the slant height (QR) of the pyramid using the Pythagorean theorem. In the right triangle \( \triangle QPR \), where \( PR = 5 \, \text{cm} \) (half of the base side), and the height of the pyramid is 12 cm:
\[
QR = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \, \text{cm}
\]
Now, calculate the Curved Surface Area (C.S.A) of the pyramid using the formula:
\[
\text{C.S.A} = \frac{1}{2} \times \text{Base Perimeter} \times \text{Slant Height}
\]
The perimeter of the base is:
\[
\text{Base Perimeter} = 4 \times \text{side} = 4 \times 10 = 40 \, \text{cm}
\]
Substitute the values into the formula:
\[
\text{C.S.A} = \frac{1}{2} \times 40 \times 13 = 260 \, \text{cm}^2
\]
Finally, the Total Surface Area (T.S.A) of the pyramid is:
\[
\text{T.S.A} = \text{C.S.A} + \text{Base Area} = 260 + 100 = 360 \, \text{cm}^2
\]
Final Answer: The correct answer is (c) 360 cm\(^2\).
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