4.6
Surface Area
Total and Lateral Surface Area
LEARNING GOALS
In this lesson you will:
t Apply formulas for the total surface area of
prisms, pyramids, cones, cylinders, and
spheres to solve problems.
t Apply the formulas for the lateral surface
area of prisms, pyramids, cones, cylinders,
and spheres to solve problems.
t Apply the formula for the area of regular
polygons to solve problems.
t lateral face
t lateral surface area
t total surface area
t apothem
I
n football, a lateral pass happens when a player passes the ball to the side or
backward, instead of forward—in a direction parallel to the goal line or away from
the goal line.
The word lateral, meaning “side,” shows up a lot in math and in other contexts. An
equilateral triangle is a triangle with equal-length sides. Bilateral talks are discussions
that take place often between two opposing sides of a conflict.
© Carnegie Learning
Discuss with students
the meaning of the word
lateral. Ask students to
think of other words that
use the same Latin stem
latus, meaning side. In
this case, students could
think of lateral surface
area as the area of the
sides (or walls) of a room.
This means that the base
needs to be well defined,
and orientation is
important. Have students
consider rectangular
prisms if the base has
not been defined and
how they would decide
what makes up the lateral
surface area.
KEY TERMS
387
4.6
Total and Lateral Surface Area
3
Problem 1
Students sketch nets of right
prisms, right pyramids, and
right cylinders in order to
determine formulas for their
lateral and total surface areas.
These formulas are then applied
to solve problems.
PROBLEM 1
Cylinders, Prisms, and Pyramids
In this lesson, you will identify and apply surface area formulas to solve problems. Let’s start
by reviewing some of these formulas.
A lateral face of a three-dimensional figure is a face that is not a base. The lateral surface
area of a three-dimensional figure is the sum of the areas of its lateral faces. The total
surface area of a three-dimensional figure is the sum of the areas of its bases and
lateral faces.
1. What units are used to describe lateral and total surface area? Explain.
Grouping
Both lateral and total surface area are described in square units, because area is
described in square units.
Discuss the definitions as a
class. Have students complete
Questions 1 and 2 with a
partner. Then have students
share their responses
as a class.
2. Consider the right rectangular prism shown. Its bases are shaded.
h
w
O
a. Sketch the bases and lateral faces of the prism. Include the dimensions of each.
Bases:
w
O
Guiding Questions
for Share Phase,
Questions 1 and 2
t Is a cube a
4
rectangular prism?
Lateral faces:
h
w
w
O
h
h
w
O
h
O
b. Determine the area of each face.
Bases: ℓw, ℓw
Lateral faces: wh, wh, ℓh, ℓh
t How can you write the
formula for the total and
lateral surface area
of a cube?
c. Use your sketch to write the formulas for the total
surface area and lateral surface area of the right
rectangular prism. Explain your reasoning.
The total surface area is the area of all of the
faces of the prism:
SA 5 2ℓw 1 2wh 1 2ℓh.
The lateral surface area is the area of all of the
faces of the prism except the bases:
SA 5 2wh 1 2ℓh.
388
Chapter 4
Three-Dimensional Figures
For right
rectangular prisms, you can
call any pairs of opposite faces “bases.”
Grouping
Discuss Question 3 as a class.
Have students complete
Questions 4 and 5 with a
partner. Then have students
share their responses
as a class.
?
3. David says that the lateral surface area of a right
rectangular prism can change, depending on what the
bases of the prism are. He calculates 3 different lateral
surface areas, L, for the prism shown.
L1 5 36 ft2
L2 5 60 ft2
L3 5 48 ft2
6 feet
2 feet
Is David correct? Explain your reasoning.
David is correct.
Guiding Question
for Share Phase,
Questions 4 and 5
t Do you prefer to use the
formula 2B 1 L or the
more specific formula for
the surface area of a
prism? Why?
3 feet
There are 3 different formulas for the lateral surface area
of a right rectangular prism, depending on which faces are considered bases:
2wh 1 2ℓh, 2ℓw 1 2ℓh, and 2ℓw 1 2wh.
The possible bases are the top and bottom faces, the front and back faces, or the two
side faces.
When applied to the prism shown, each of these formulas gives a different lateral
surface area for the prism.
4. Determine the lateral surface area and total
surface area of the right prism shown.
The bases of the prism are shaded.
4 centimeters
4 centimeters
22 centimeters
a. Identify the length, width, and height of
the prism.
Let ℓ 5 length (22 cm), w 5 width (4 cm), and h 5 height (4 cm).
b. Apply the formula to determine the lateral surface area of the prism.
The lateral surface area of the prism is 352 square centimeters.
Lateral surface area 5 2ℓw 1 2ℓh
5 2(22)(4) 1 2(22)(4)
5 352
c. Apply the formula to determine the total surface area of the prism.
The total surface area of the prism is 384 square centimeters.
Total surface area 5 2ℓw 1 2ℓh 1 2wh
© Carnegie Learning
5 2(22)(4) 1 2(22)(4) 1 2(4)(4)
5 384
4.6
Total and Lateral Surface Area
3
Grouping
Have students complete
Question 6 with a partner.
Then have students share their
responses as a class.
5. Explain why Vicki is correct.
The net of any prism includes 2 congruent
bases and a number of lateral faces. The
total area of these faces represents the
total surface area of the prism.
Vicki
One formula for the total surface
area of any prism can be written
as 2B + L, where B represents the
area of each base and L represents
the lateral surface area.
Guiding Question
for Share Phase,
Question 6
t How can you use the formula
for the perimeter—2l 1 2w—
to show that Michael
is correct?
6. Consider the right rectangular pyramid shown.
a. Sketch the bases and lateral faces of the pyramid. Include
the dimensions.
Base:
s
Lateral Faces:
w
s
O
w
O
s
O
O
s
w
s
w
4
b. Determine the area of each face.
Area of base: ℓw
__ __ __ __
Area of lateral faces: 1 ℓs, 1 ℓs, 1 ws, 1 ws
2
2
2
2
c. Use your sketch to write the formulas for the total surface area and lateral surface
area of the pyramid. Explain your reasoning.
Let ℓ 5 length of base, w 5 width of base, and s 5 slant height.
The total surface area is the area of all of the faces of the pyramid:
1sw) 1 2(__
1 ℓs)
SA 5 ℓw 1 2(__
2
2
5 ℓw 1 sw 1 ℓs
The lateral surface area is the area of all of the faces of the pyramid except
the bases: SA 5 sw 1 ℓs.
390
Chapter 4
Three-Dimensional Figures
Grouping
Have students complete
Question 7 with a partner.
Then have students share their
responses as a class.
Guiding Questions
for Share Phase,
Question 7
t What units are used to
describe lateral and total
surface area?
t How can you check that your
answers are reasonable?
d. Use your sketch and the formula you
determined to explain why Michael is
correct.
Michael
Answers will vary.
Lateral surface area of a right
1
rectangular pyramid 5 _Ps.
2
Total surface area of a
I determined that the formula for the
lateral surface area of a rectangular
pyramid is sw 1 ℓs.
right rectangular pyramid
1
5 _Ps + B.
2
P 5 perimeter of base,
s 5 slant height, and
B 5 area of base.
__
1s(2ℓ 1 2w), or 1 Ps,
This is equivalent to __
2
2
where P, or 2ℓ 1 2w, is the perimeter of
the base.
For the total surface area, I added the
area of the base, B to the lateral surface
area, 1 Ps.
2
__
__1 Ps 1 B
2
7. Determine the lateral surface area and total surface area of the right
rectangular pyramid shown.
a. Apply the formula to determine the lateral surface area of the
pyramid.
The lateral surface area of the pyramid is 32 square inches.
4 in.
3 in.
5 in.
1 (4)(2 3 5 1 2 3 3)
Lateral surface area 5 __
2
5 32
b. Apply the formula to determine the total surface area of the pyramid.
The total surface area of the pyramid is 47 square inches.
© Carnegie Learning
1 (4)(2 3 5 1 2 3 3) 1 (5 3 3)
Total surface area 5 __
2
5 47
4.6
Total and Lateral Surface Area
3
Grouping
Have students complete
Question 8 with a partner.
Then have students share their
responses as a class.
Guiding Questions
for Share Phase,
Question 8
t What is the formula for the
8. Consider the right cylinder shown.
r
a. Sketch the bases and lateral faces of the cylinder. Include the
dimensions.
Bases:
h
Lateral Face:
r
r
h
2pr
b. Determine the area of each face.
Area of bases: pr2, pr2
area of a circle?
t What shape is the lateral face
Area of lateral face: 2prh
of the cylinder?
t How is the circumference of
the base related to the lateral
face of the cylinder?
c. Use your sketch to write the formulas for the total
surface area and lateral surface area of the cylinder.
Explain your reasoning.
Recall that the
width of the lateral face of
a cylinder is equal to the
circumference of the
base.
Let h 5 height of cylinder and r 5 radius of cylinder.
The total surface area is the area of all of the faces
of the cylinder: Total SA 5 2pr2 + 2prh
Grouping
Have students complete
Question 9 with a partner.
Then have students share their
responses as a class.
4
The lateral surface area is the area of all
of the faces of the cylinder except the
bases: Lateral SA 5 2prh.
9. A cylindrical paint roller has a diameter of 2.5 inches and a length of 10 inches.
a. Apply the formula to determine the lateral surface area of the paint roller.
The lateral surface area is 2p(1.25)(10), or approximately 78.54 square inches.
Guiding Questions
for Share Phase,
Question 9
t What is the radius of the
paint roller?
t Which measure—lateral
surface area or total surface
area—would matter when
painting with the paint roller?
t How can you check the
reasonableness of
your answers?
392
Chapter 4
Three-Dimensional Figures
b. Apply the formula to determine the total surface area of the paint roller.
The total surface area is 2p(1.25)(10) 1 2p(1.25)2, or approximately
88.36 square inches.
Problem 2
Students investigate and apply
surface area formulas for solid
figures with regular polygons
as bases.
Grouping
Discuss the formula for the
area of a regular polygon and
the definition of apothem as a
class. Have students complete
Questions 1 and 2 with a
partner. Then have students
share their responses as
a class.
Guiding Questions
for Share Phase,
Questions 1 and 2
t Which segment length in
the diagram represents
the apothem?
t Would you need the apothem
to determine the lateral
surface area of the pyramid?
Why or why not?
t How can you use formulas
A Regular Problem
You have learned previously that the formula for the area
of a regular polygon—a polygon with all congruent
1 Pa, where a represents the length
sides—is A 5 __
2
of the apothem and P represents the perimeter of
the polygon.
You can apply this formula to solve problems involving
surface area.
Recall that
the apothem is the
length of a line segment from
the center of the polygon to
the midpoint of
a side.
1. Consider the right hexagonal pyramid shown.
Its base is a regular hexagon.
14 ft
8 ft
6 ft
a. What formula is used to determine the total surface area of the pyramid?
1Ps 1 B describes the total surface area of the pyramid, where P
The formula __
2
represents the perimeter of the base, s represents the slant height, and B
represents the area of the base.
b. Apply the formula for the area of a regular polygon to determine the area of the base
of the hexagonal pyramid. Show your work.
The area of the base of the hexagonal pyramid is 144 square feet.
1 (6 ft)(8 ft 3 6)
B 5 __
2
1 (288 ft2)
5 __
2
5 144
c. Determine the total surface area of the hexagonal pyramid.
The total surface area of the pyramid is 480 square feet.
The area of the base, B, is 144 square feet.
The perimeter of the base, P, is 8 ft 3 6, or 48 feet.
The slant height is 14 feet.
1(48 ft)(14 ft) 1 144 ft2 5 480 ft2
__
2
© Carnegie Learning
you have learned previously
to verify the lateral and
total surface area of the
right hexagonal prism?
PROBLEM 2
4.6
Total and Lateral Surface Area
3
2. The right prism shown has shaded bases that are regular
polygons. Apply the formulas you know to determine the total
and lateral surface area of the prism.
Total surface area 5 2920 cm2
68 cm
Lateral surface area 5 2720 cm2
The formula for the total surface area of a prism can be
written as 2B 1 L.
1 Pa, the area of a regular polygon, for B in
I can substitute __
2
this formula because each base is a regular pentagon:
1
2(__Pa) 1 L.
2
P 5 8 cm 3 5 5 40 cm
8 cm
5 cm
a 5 5 cm
L 5 5(68 cm 3 8 cm) 5 2720 cm2
1 3 40 3 5) 1 2720,
The total surface area of the pentagonal prism is 2(__
2
or 2920 square centimeters.
The lateral surface area of the pentagonal prism is equal to L, or
2720 square centimeters.
Problem 3
PROBLEM 3
Students investigate formulas
for the total and lateral
surface areas of cones and
spheres. Students conclude by
summarizing the surface area
formulas they have learned in
this lesson.
4
Cones and Spheres
You can also apply the formulas for the lateral and total surface areas of cones and spheres
to solve problems.
A right cone is made up of two faces—a circular base and a wedge-shaped lateral face.
s
r
Base:
Lateral Face:
r
s
Grouping
Discuss the net of a cone as a
class. Have students complete
Questions 1 and 2 with a
partner. Then have students
share their responses as a
class. Discuss Question 3
as a class.
394
Chapter 4
Three-Dimensional Figures
2pr
The area of the circular base is given by pr2.
1(2pr)(s), or prs, where s is the slant height of the cone.
The area of the lateral face is given by __
2
Guiding Questions
for Share Phase,
Questions 1 and 2
t Describe how the total and
lateral surface area of a cone
is similar to the surface areas
of other solid figures.
t How can you check the
reasonableness of
your answers?
1. Write the formulas for the lateral surface area and total surface area of a right cone.
1(2pr)(s), or prs.
The lateral surface area of a cone is given by __
2
I add the area of the circular base, pr2, to determine the total surface area,
which is pr2 1 prs.
6.4 feet
4 feet
2. Determine the lateral and total surface area of the cone. Round to
the nearest hundredth.
a. Determine the slant height, s, and the radius, r, of the cone.
s 5 6.4 ft
10 feet
r 5 5 ft
b. Apply the formula to determine the lateral surface area of the cone.
The lateral surface area of the cone is approximately 100.53 square feet.
p(5)(6.4) 5 32p
¯ 100.53
c. Apply the formula to determine the total surface area of the cone.
The total surface area of the cone is approximately 100.53 square feet.
p(5)2 1 p(5)(6.4) 5 25p 1 32p
5 57p
¯ 179.07
?
3. Benjamin argued that he could increase the total surface area of a cone without
increasing the radius of the base.
Is Benjamin correct? Use the lateral and total surface area formulas to explain your
reasoning.
Benjamin is correct. He can increase the total surface area by increasing the slant
height of the cone.
© Carnegie Learning
Given the formula for total surface area of a cone, pr2 1 prs, increasing the slant
height increases the lateral surface area, prs, and total surface area but does not
change the area of the circular base, pr2.
4.6
Total and Lateral Surface Area
3
Grouping
Discuss the surface area of a
sphere and complete Question
4 as a class. Have students
complete Question 5 with a
partner. Then have students
share their responses as
a class.
You can think of a sphere as a solid figure with
bases that are points. Each of these points has an
area of 0, so the total surface area of a sphere is
equal to its lateral surface area.
base area 5 0 in.2
r 5 5.2 in.
total surface area 5 lateral surface area
5 4pr2
base area 5 0 in.2
4. Apply the formula to determine the total and
lateral surface area of the sphere shown.
The surface area of the sphere is approximately 339.79 square inches.
Guiding Questions
for Share Phase,
Question 5
t Which surface area formulas
4p(5.2)2 ¯ 339.79
5. Complete the table to record the formulas for the lateral surface area and total surface
area of the figures you studied in this lesson. Identify what the variables in your formulas
represent.
involve slant height?
Surface Area Formulas
t What would the slant height
of a right cone be if its lateral
surface area were equal to
the lateral surface area of a
right cylinder?
Figure
4
Total Surface Area
2ℓw 1 2ℓh 1 2wh, or 2B 1 L
2ℓh 1 2wh
Right Rectangular Prism
t Describe the height of a
right cylinder when its lateral
surface area is equal to the
surface area of a sphere.
Lateral Surface Area
ℓ 5 length
w 5 width
h 5 height
1Ps
__
Right Rectangular Pyramid
2
P 5 perimeter of base
s 5 slant height
ℓ 5 length
b 5 width
c 5 height
B 5 area of base
L 5 lateral surface area
1 Ps 1 B, or B 1 L
__
2
P 5 perimeter of base
s 5 slant height
B 5 area of base
L 5 lateral surface area
2pr2 1 2prh, or 2B 1 L
2prh
Right Cylinder
r 5 radius of cylinder
h 5 height of cylinder
r 5 radius of cylinder
h 5 height of cylinder
B 5 area of base
L 5 lateral surface area
pr2 1 prs, or B 1 L
prs
Right Cone
r 5 radius of cone
s 5 slant height
4pr2
r 5 radius of cone
s 5 slant height
B 5 area of base
L 5 lateral surface area
4pr2
Sphere
r 5 radius of sphere
396
Chapter 4
Three-Dimensional Figures
r 5 radius of sphere