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ACT Math : How to find the surface area of a pyramid

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Example Question #1 : 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:

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

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

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

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

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

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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Example Question #2 : How To Find The Surface Area Of A Pyramid

Consider a square-base pyramid, . If one of the triangular faces of is analyzed individually and it is found to have a base of and a height of . What is the total surface area of ?

Possible Answers:

Correct answer:

Explanation:

The surface area of this pyramid is equal to the area of the square base plus the areas of each of the four triangular sides.

We are given the base and height of the triangular faces.

We can use $a=\frac{b \cdot h}{2}$ and plug in 3 and 6, giving us $a=\frac{3\cdot 6}{2}=9$ .

Because the base of the triangular faces is equal to 3, we know the area of the square base is $3\cdot 3=9$ .

Therefore, our total surface area is equal to 4(9)+9=45

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Example Question #3 : How To Find The Surface Area Of A Pyramid

The largest of the Great Egyptian Pyramids has a base length and width of 230 metrers and a height of 147 meters. What is the surface area of this rectangular pyramid in scientific notation?

Possible Answers:

$7.7\times 10^{4}\ \textup{m}^{2}$

$1.4\times 10^{5}\ \textup{m}^{2}$

$2.3\times 10^{5}\ \textup{m}^{2}$

$3.2\times 10^{5}\ \textup{m}^{2}$

$1.2\times 10^{6}\ \textup{m}^{2}$

Correct answer:

$1.4\times 10^{5}\ \textup{m}^{2}$

Explanation:

The equation for surface area of a rectangular pyramid is as follows:

$SA=l\cdot {w}+l\sqrt{\left ( \frac{w}{2}\right )^{2}+h^{2}}+w\sqrt{\left ( \frac{l}{2} \right )^{2}+h^{2}}$

Plug in the values for length, width and height:

$SA=230\cdot {230}+230\sqrt{\left ( \frac{230}{2}\right )^{2}+147^{2}}+230\sqrt{\left ( \frac{230}{2} \right )^{2}+147^{2}}$

$SA=52900+230\sqrt{115^{2}+147^{2}}+230\sqrt{115^{2}+147^{2}}$

$SA=52900+2\cdot (230\sqrt{115^{2}+147^{2}})$

$SA=52900+2\cdot (230\sqrt{13225+21609})$

$SA=52900+2\cdot (42926.9)$

$SA=138753.8 \approx 1.4\cdot10^{5}m^{2}$

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Example Question #4 : How To Find The Surface Area Of A Pyramid

Find the surface of a isoceles pyramid whose height is , base side length is , and slant height is .

Possible Answers:

Correct answer:

Explanation:

To find surface area of a pyramid, simply find the surface area of each of the faces and add them together.

$\textup{Base}=s^2=6^2=36$

$\textup{Triangle face}=\frac{1}{2}bh=\frac{1}{2}*6*4=12$

Since there are 4 trangular faces, we have to multiply that surface area by 4. Thus,

SA=36+4*12=36+48=84

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ACT Math : How to find the surface area of a pyramid

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : Pyramids

Example Question #2 : How To Find The Surface Area Of A Pyramid

Example Question #3 : How To Find The Surface Area Of A Pyramid

Example Question #4 : How To Find The Surface Area Of A Pyramid

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