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High School Math : Pyramids

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Example Question #11 : Pyramids

What is the surface area of a square pyramid with a height of 12 in and a base side length of 10 in?

Possible Answers:

$260 \textup{ in}^{2}$

$100 \textup{ in}^{2}$

$150 \textup{ in}^{2}$

$360 \textup{ in}^{2}$

$420 \textup{ in}^{2}$

Correct answer:

$360 \textup{ in}^{2}$

Explanation:

The surface area of a square pyramid can be broken into the area of the square base and the areas of the four triangluar sides. The area of a square is given by:

$A = s^{2} = 100 in^{2}$

The area of a triangle is:

$A = \frac{1}{2} bh$

The given height of 12 in is from the vertex to the center of the base. We need to calculate the slant height of the triangular face by using the Pythagorean Theorem:

$SH^{2} = H^{2} + B^{2}$

where H=12 and B= 5 (half the base side) resulting in a slant height of 13 in.

So, the area of the triangle is:

$A = \frac{1}{2} bh = \frac{1}{2} \cdot 10 \cdot 13 = 65 in^{2}$

There are four triangular sides totaling $260 in^{2}$ for the sides.

The total surface area is thus $360 in^{2}$ , including all four sides and the base.

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High School Math : Pyramids

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Example Question #11 : Pyramids

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