Surface Area of Pyramids
Bonnie Niemann
Jen Kershaw
Brenda Meery
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both in
the U.S. and worldwide. Using an open-source, collaborative, and
web-based compilation model, CK-12 pioneers and promotes the
creation and distribution of high-quality, adaptive online textbooks
that can be mixed, modified and printed (i.e., the FlexBook®
textbooks).
Copyright © 2016 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including CK-12
Curriculum Material) is made available to Users in accordance
with the Creative Commons Attribution-Non-Commercial 3.0
Unported (CC BY-NC 3.0) License (http://creativecommons.org/
licenses/by-nc/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated
herein by this reference.
Complete terms can be found at http://www.ck12.org/about/
terms-of-use.
Printed: May 5, 2016
AUTHORS
Bonnie Niemann
Jen Kershaw
Brenda Meery
www.ck12.org
C HAPTER
Chapter 1. Surface Area of Pyramids
1
Surface Area of Pyramids
In this concept, you will learn how to find the surface area of a pyramid.
Let’s Think About It
Janice and Julian want to make a tent that is in the shape of a square pyramid for their summer camping adventure.
They want it to have a base of 6 feet wide and a slant height of 8 feet . They need to find out the surface area of the
tent so they know how much canvas to buy. How can they calculate the surface area of their future tent?
In this concept, you will learn how to find the surface area of a pyramid.
Guidance
A pyramid has sides that are triangular faces and a base. The base can be any shape. Here are three examples:
The surface area is the sum of all of the areas of the faces in a solid figure. Imagine wrapping a pyramid in wrapping
paper, like a present. The amount of wrapping paper needed to cover the figure represents its surface area. To find
the surface area, you must be able to calculate the area of each face and then add these areas together.
One way to find the surface area of a three-dimensional figure is by using a net. A net is a two-dimensional diagram
of a three-dimensional figure. Imagine unfolding a pyramid so that it is completely flat. It would look something
like this net.
1
www.ck12.org
With a net, you can see each face of the pyramid more clearly. To find the surface area, you need to calculate the
area for each face of the net, the sides and the base.
The side faces of a pyramid are always triangles, so the area formula for triangles is used to calculate their areas:
A = 21 bh.
The area of the base depends on what shape it is. Remember, pyramids can have bases in the shape of a triangle,
square, rectangle, or any other polygon. Use whichever area formula is appropriate for the shape. Here are some
common formulas to find area.
rectangle : A = lw
square : A = s2
1
triangle : A = bh
2
Take a look at the pyramid below. Which formula should you use to find the area of the base?
The base of the pyramid is a square with sides of 6 centimeters, so you should use the area formula for a square:
A = s2 . Now find the area of the base.
A = 62
A = 36 sq.cm
Next, find the area of one of the triangles. You use the formula A = 12 bh. The base width is 6 cm and the slant height,
the height of the side is 4 cm.
2
www.ck12.org
Chapter 1. Surface Area of Pyramids
1
A = (6)(4)
2
1
A = (24)
2
A = 12 sq.cm
There are four triangles, so multiply this number by four and then add it to the area of the square.
SA = 4(12) + 36
SA = 48 + 36
SA = 84 sq.cm
Nets let you see each face of a pyramid so that you can calculate its area. However, you can also use a formula to
represent the surface area of a pyramid. The formula is like a short cut because you can put all of the measurements
in for the appropriate variable in the formula and solve for SA, surface area.
Here is the formula for finding the surface area of a pyramid.
SA =
1
perimeter × slant height + B
2
The first part of the formula, 21 perimeter × slant height, is a quick way of finding the area of all of the triangular
sides of the pyramid at once. Remember, the area formula for a triangle is A = 12 bh. In the formula, b stands for base.
The perimeter of the pyramid’s bottom face represents all of the bases of the triangular faces at once, because it’s
their sum. The height of each triangle is always the same; this is called the slant height of the pyramid. Therefore
“ 12 perimeter × slant height” is really the same as 12 bh.
The B in the formula represents the area of the base. Remember, pyramids can have bases of different shapes, so the
area formula is used to find B varies. You find the area of the base first and then put it into the formula in place of B.
Let’s look at an example.
What is the surface area of the pyramid below?
This is a square pyramid. The four sides of the base are all 8 inches, so the perimeter of the base is 8 × 4 = 32 inches.
You will need to use the area formula for a square to find B.
B = s2
B = (8)2
B = 64 in.2
3
www.ck12.org
Now you have all the information that you need. Put it into the formula and solve for SA, surface area.
1
perimeter × slant height + B
2
1
SA = (32) × 3 + 64
2
SA = (16 × 3) + 64
SA =
SA = 48 + 64
SA = 112 in.2
The answer is the surface area of the pyramid is 112 square inches.
Guided Practice
Find the surface area of the figure below.
First, calculate the area of the base and the perimeter. To do this, identify the shape of the base and use the
appropriate formula for B . This is a triangular pyramid because its base is a triangle. That means you need to use
the area formula for triangles to find B.
1
B = bh
2
1
B = (16)(13.86)
2
B = 8(13.86)
B = 110.88 cm2
The sides of the base are all the same length, so you can calculate the perimeter by multiplying 16 × 3 = 48.
Next, plug the values into the formula for surface area of a pyramid and multiply the values together inside the
brackets:
1
perimeter × slant height + B
2
1
SA = (48) × 6 + 110.88
2
SA =
4
www.ck12.org
Chapter 1. Surface Area of Pyramids
SA = (24 × 6) + 110.88
SA = 144 + 110.88
Then, add the values together, making sure to include the appropriate unit of measurement:
SA = 144 + 110.88
SA = 254.88 cm2
The answer is the surface area of this triangular pyramid is 254.88 square centimeters.
Examples
Example 1
Find the surface area of a square pyramid with side of 6 cm, slant height of 5 cm.
First, calculate the area and perimeter of the square base:
B = s2
B = 62
B = 36cm2
P = 4× 6
P = 24 cm
Next, plug the values into the formula for surface area of a pyramid and multiply the values together inside the
brackets:
1
perimeter × slant height + B
2
1
SA = (24) × 5 + 36
2
SA =
SA = (12 × 5) + 36
SA = 60 + 36
5
www.ck12.org
Then, add the values together for the answer, remembering the appropriate unit of measurement.
SA = 60 + 36
SA = 96 cm2
The answer is the surface area of the square pyramid is 96 cm2 .
Example 2
Find the surface area of a rectangular pyramid with a length of 6 in, a width of 4 in and a slant height of 3 in.
First, calculate the area and perimeter of the rectangular base:
B = lw
B = 6×4
B = 24 in2
P = 2l + 2w
P = 2(6) + 2(4)
P = 12 + 8
P = 20 in
Next, plug the values into the formula for surface area of a pyramid and multiply the values together inside the
brackets:
1
perimeter × slant height + B
2
1
SA = (20) × 3 + 24
2
SA =
SA = (10 × 3) + 24
SA = 30 + 24
Then, add the values together for the answer. Remember the appropriate unit of measurement.
SA = 30 + 24
SA = 54 in2
The answer is 54 in2 .
6
www.ck12.org
Chapter 1. Surface Area of Pyramids
Example 3
Find the surface area of a square pyramid with side of 8 in and slant height of 9 in.
First, calculate the area and perimeter of the square base.
B = s2
B = 82
B = 64in2
P = 4× 8
P = 32 in
Next, plug the values into the formula for surface area of a pyramid and multiply the values together inside the
brackets:
1
perimeter × slant height + B
2
1
SA = (32) × 9 + 64
2
SA =
SA = (16 × 9) + 64
SA = 144 + 64
Then, add the values together for the answer, remembering the appropriate unit of measurement:
SA = 144 + 64
SA = 208 in2
The answer is the surface area of the square pyramid is 208 in2 .
7
www.ck12.org
Follow Up
Remember Janice and Julian and their square pyramid tent?
They are planning for their tent to have a base of 6 feet wide and a slant height of 8 feet. How much canvas will
they need?
First, calculate the area and perimeter of the square base:
B = s2
B = 62
B = 36 f t 2
P = 4× 6
P = 24 f t
Next, plug the values into the formula for surface area of a pyramid and multiply the values together inside the
brackets:
1
perimeter × slant height + B
2
1
SA = (24) × 8 + 36
2
SA =
8
www.ck12.org
Chapter 1. Surface Area of Pyramids
SA = (12 × 8) + 36
SA = 96 + 36
Then, add the values together for the answer, making sure to include the appropriate unit of measurement:
SA = 96 + 36
SA = 132 f t 2
The answer is the surface area of the tent is 132 square feet. Janice and Julian will require 132 square feet of canvas
to make their tent.
Explore More
Find the surface area of each square pyramid. Remember that b means base and sh means slant height.
1. b = 4 in, sh = 5 in
2. b = 4 in, sh = 6 in
3. b = 6 in, sh = 8 in
4. b = 5 in, sh = 7 in
5. b = 7 m, sh = 9 m
6. b = 8 m, sh = 10 m
7. b = 9 cm, sh = 11 cm
8. b = 11 m, sh = 13 m
9. b = 6 in, sh = 9 in
10. b = 10 cm, sh = 12 cm
Find the surface area of each rectangular pyramid.
11. l = 10 in, w = 8 in, sh = 6 in
12. l = 12 f t, w = 8 f t, sh = 10 f t
13. l = 4 in, w = 2 in, sh = 3 in
14. l = 16 m, w = 12 m, sh = 11 m
15. l = 15 in, w = 10 in, sh = 12 in
References
1. butforthesky.com. https://flic.kr/p/agdkG4 .
2. butforthesky.com. https://flic.kr/p/agdkG4 .
9