2 Surface Areas of Prisms and Cylinders Mafhematics Florida Standards MAFS.912.G-MG.1.1 Use geometric shapes, their measures, and their properties to describe objects. MP 1, MR 3, MR 4, MR 6, MR 7, MR 8 Objective To find the surface area of a prism and a cylinder Getting Ready! m A piece of string is wrapped once around an empty paper towel tube. The ends of the 2 m string are attached to each end of the tube OS shown. How long is the piece of string? Justify your reasoning. Vou II flatten this problem out in no time! MATHEMATICAL PRACTICES Lesson Vocabulary prism (base, lateral face, altitude, height, lateral area, surface area) right prism oblique prism cylinder (base, altitude, height, lateral area, surface area) right cylinder oblique cylinder In the Solve It, you investigated the structure of a tube. In this lesson, you will learn properties of three-dimensional figures by investigating their surfaces. Essential Understanding To find the surface area of a three-dimensional figure, find the sum of the areas of all the surfaces of the figure. A prism is a polyhedron with two congruent, parallel faces, called bases. The other faces are lateral faces. You can name a prism using the shape of its bases. Bases Bases Latera Latera Pentagonal prism Triangular prism An altitude of a prism is a perpendicular segment that joins the planes of the bases. The height /; of a prism is the length of an altitude. A prism may either be right or oblique. ; Right prisms Oblique prisms In a right prism, the lateral faces are rectangles and a lateral edge is an altitude. In an oblique prism, some or all of the lateral faces are nonrectangular. In this book, you may assume tliat a prism is a right prism unless stated or pictured otherwise. C PovyerGeometry.com { Lesson 11-2 Surface Areas of Prisms and Cylinders 699
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2 Surface Areas of Prismsand Cylinders
Mafhematics Florida Standards
MAFS.912.G-MG.1.1 Use geometric shapes,their measures, and their properties to describeobjects.
MP 1, MR 3, MR 4, MR 6, MR 7, MR 8
Objective To find the surface area of a prism and a cylinder
Getting Ready!
m
A piece of string is wrapped once around anempty paper towel tube. The ends of the 2 mstring are attached to each end of the tubeOS shown. How long is the piece of string?Justify your reasoning.
In the Solve It, you investigated the structure of a tube. In this lesson, you will learn
properties of three-dimensional figures by investigating their surfaces.
Essential Understanding To find the surface area of a three-dimensional figure,find the sum of the areas of all the surfaces of the figure.
A prism is a polyhedron with two congruent, parallel faces, called bases. The other faces
are lateral faces. You can name a prism using the shape of its bases.
Bases Bases
Latera
Latera
Pentagonal prism Triangular prism
An altitude of a prism is a perpendicular segment that joins the planes of the bases. Theheight /; of a prism is the length of an altitude. A prism may either be right or oblique.
;
Right prisms Oblique prisms
In a right prism, the lateral faces are rectangles and a lateral edge is an altitude. In anoblique prism, some or all of the lateral faces are nonrectangular. In this book, you mayassume tliat a prism is a right prism unless stated or pictured otherwise.
C PovyerGeometry.com { Lesson 11-2 Surface Areas of Prisms and Cylinders 699
How do you knowwhat units to use?
In the prism, therectangle marked 5on one side and 4 on
the other has area
5 cm X 4 cm, or 20 cm^.So use cm^ as the unitfor the surface area of
the prism.
The lateral area (L.A.) of a prism is the sum of the areas of the lateral faces. Thesurface area (S.A.) is the sum of the lateral area and the area of the two bases.
Problem 1 Using a Net to Find Surface Area of a Prism
What is the surface area of the prism at the right? Use a net.
Draw a net for the prism. Then calculate the surface area.
S.A. = sum of areas of all the faces
=5*4+5*3+5*4+5*3+3*4+3*4
= 20+15 + 20+ 15 + 12+12
= 94
5 cm>-•
4 cm
The surface area of the prism is 94 cm^.
Got It? 1. What is the surface area of the triangular prism?Use a net.
5 cm
You can find formulas for lateral and surface areas of a prism by using a net.
d
)
, ha'
b c
kh
The perimeter pof a base is
a+b+c+d.
a a c
a b c d
B ̂.. .
3 cm
Lateral Area = ph
You can use the formulas with any right prism.
Surface Area = L.A. + 26
Theorem 11-1 Lateral and Surface Areas of a Prism
The lateral area of a right prism is the product of the
perimeter of the base and the height of the prism.L.A. = ph
The surface area of a right prism is the sum of the lateral ^area and the areas of the two bases.
S.A. = L.A. + 26
p is the perimeterof a base.
;—
B is the area of a base.
700 Chapter 11 Surface Area and Volume
What do you need tofind first?
You first need to find
the missing side lengthof a triangular base sothat you can find theperimeter of a base.
Using Formulas to Find Surface Area of a Prism
What is the surface area of the prism at the right? 3 cm
Step 1 Find the perimeter of a base.
The perimeter of the base is the sum of the side lengths of thetriangle. Since the base is a right triangle, the hypotenuse is
cm, or 5 cm, by the Pythagorean Theorem.
p = 3 + 4 + 5 = 12
Step 2 Find the lateral area of the prism.
L.A. = ph Use the formula for lateral area.
= 12 • 6 Substitute 12 for p and 6 for h.
= 72 Simplify.
Find the area of a base.Step 3
4 cm
6 cm
Which height do youneed?
For problems involvingsolids, make it a habitto note which heightthe formula requires.In Step 3, you need theheight of the triangle, notthe height of the prism.
Use the formula for the area of a triangle.
= |(3 • 4) Substitute 3 for b and 4 for h.= 6
Step 4 Find the surface area of the prism.
S.A. = L.A. + 2B Use the formula for surface area.
= 72 + 2(6} Substitute 72 for L.A. and 6 for B.
= 84 Simplify.
The surface area of the prism is 84 cm^.
^ Got It? 2. a. What is the lateral area of the prism at the right?b. What is the area of a base in simplest radical form?c. What is the surface area of the prism rounded to a
whole number?
\
A cylinder is a solid that has two congruent parallel bases that are circles. An altitude of
a cylinder is a perpendicular segment that joins the planes of the bases. The height h ofa cylinder is the length of an altitude.
Bases
Right cylinders Oblique cylinders
In a right cylinder, the segment joining the centers of the bases is an altitude. In anoblique cylinder, the segment joining the centers is not perpendicular to the planescontaining the bases. In this book, you may assume that a cylinder is a right cylinderunless stated or pictured otherwise.
crrbwerGeometry.com Lesson 11-2 Surface Areas of Prisms and Cylinders 701
To find the area of the curved surface of a cylinder, visualize "unrolling" it. The area ofthe resulting rectangle is the lateral area of the cylinder. The surface area of a cylinderis the sum of the lateral area and the areas of the two circular bases. You can find
formulas for these areas by looking at a net for a cylinder.
£3:27rr
oThe circumference
of a base is lirr.
The area 6 of a base
IS Trr.
Lateral Area = lirrh Surface Area = L.A. + lirr^
How is finding thesurface area of a
cylinder like findingthe surface area of a
prism?For both, you need tofind the L.A. and add it to
twice the area of a base.
Theorem 11-2 Lateral and Surface Areas of a Cylinder
The lateral area of a right cylinder is the product of thecircumference of the base and the height of the cylinder.
L.A. = 27rr • h, or L.A. = irdh
The surface area of a right cylinder is the sum of thelateral area and the areas of the two bases.
S.A. = L.A. + 2B, or S.A. = 27rrh + 27rr^ B is the area of a base.
Finding Surface Area of a Cylinder
Multiple Choice The radius of the base of a cylinder is 4 in. and its height is 6 in.What is the surface area of the cylinder in terms of nl
CA^ 32'7r in.^ CX> 427r in.^ CO BOtt in.^ CO 1207r in.^
S.A. = L.A. + 2B Use the formula for surface area of a cylinder.
= 2'TTrh + 2(7rr^) Substitute the formulas for lateral area and area of a circle.
= 27r(4)(6) + 2(774^) Substitute 4 for r and 6 for h.
= 4877 + 3277 Simplify.
= 8077
The surface area of the cylinder is BOtt in.^. The correct choice is C.
Got It? 3. A cylinder has a height of 9 cm and a radius of 10 cm. What is thesurface area of the cylinder in terms of 77?
702 Chapter 11 Surface Area and Volume
rui^h-
What is the probiemasking you to find?The area covered by onefull turn of the roller is
the rectangular part ofthe net of a cylinder. Thisis the lateral area.
Problem 4f Finding Lateral Area of a Cylinder
Interior Design You are using the cylindrical stencil roller below to paint patterns onyour floor. What area does the roller cover in one full turn?
The area covered is the lateral area of a cylinder with height 6 in. and diameter 2.5 in.
L.A. = Trdh Use the formula for lateral area of a cylinder.
= Tr(2.5)(6) Substitute 2.5 for d and 6 for h.
= IStt « 47.1 Simplify,
j In one full turn, the stencil roller covers about 47.1 in.^.
@ , ̂ Got It? 4. a. A smaller stencil roller has a height of 1.5 in. and the same diameter as theroller in Problem 4. What area does the smaller roller cover In one turn?
Round your answer to the nearest tenth,
b. Reasoning What is the ratio ofthe smaller roller's height to the largerroller's height? What is the ratio of the areas the rollers can cover in one
turn (smaller to larger)?
Lesson Check
Do you know HOW?What is the surface area of each prism?
2. 7 ft
5 in,
What is the surface area of each cylinder?
3. ^3 cm 4. 12 m
5 cm I * 10 m
Do you UNDERSTAND?
5. Vocabulary Name the lateral
faces and the bases ofthe prismat the right.
6. Error Analysis Your friend
drew a net of a cylinder. What
is your friend's error? Explain.
MATHEMATICAL
PRACTICES
2^^ 3 cm
o1
C PowerGeometry.com Lesson 11-2 Surface Areas of Prisms and Cylinders 703
Practice and Problem-Solving Exercises
Practice Use a net to find the surface area of each prism.
MATHEMATICAL
PRACTICES
t
29 cm
7.
;
^ See Problem 1.
9.
6ft
6 ft
19 cm^^6.5 cm 6 ft
4 in.
4 in.
10. a. Classify the prism at the right.
b. Find the lateral area of the prism.
c. The bases are regular hexagons. Find the sumof their areas.
d. Find the surface area of the prism.
10 cm
4 cm
Use formulas to find the surface area of each prism. Round your answer to the ^ See Problem 2.nearest whole number.
11.
j
4 ft
5 ft10 ft
5 in.
Find the lateral area of each cylinder to the nearest whole number.
14. .4 in.
S-j in.
15.
6 m 9 m
13. Regularoctagon
5 cm
^ See Problems 3 and 4.
8 cm
20 cm
Find the surface area of each cylinder in terms of tt.
17. —^2 cm 18.
8 cm 4 cm
19. 7 in.
11 in.
62) 20. Packaging A cylindrical carton of oatmeal with radius 3.5 in. is 9 in. tall. If allsurfaces except the top are made of cardboard, how much cardboard is used to
make the oatmeal carton? Assume no surfaces overlap. Round your answer to the