Surface Areas of Prisms
Objective
After studying this section, you will be able to
QFind the surface areas of prisms
Part One: Introduction
Solids with flat faces are called polyhedra (meaning "many faces").
The faces are polygons, and the lines where they intersect are called
edges.
<G>
One familiar type ofpolyhedron is the prism. Here are three
examples:
Triangle /.2l--"
•u
Triangular Prism
<
i_^ Rectangle
Rectangular Prism
•<; rV'i^r#gbriiN
Pentagonal Prism
Every prism has two congruent parallel face?
[shaded iii iiie examples) and a set of paral
lel edges that connect corresponding vertices
of the two parallel faces.
The two parallel and congruent faces are
Lateral
called bases. The parallel edges joining the
Face
Lateral
\ V^Edge
—\->
vertices of the bases are called lateral
edges. The faces of the prism that are not
bases are called lateral faces. The lateral
faces of all prisms are parallelograms.
Therefore, we name prisms by their
bases—a prism with hexagonal bases, for
example, is called a hexagonal prism.
'Base
Section 12.1
Surface Areas of Prisms
561
Definition
The lateral surface area of a prism is the sum of the
areas of the lateral faces.
Definition
The totalsurface area of a prism is the sum of the
prism's lateral area and the areas of the two bases.
If the lateral edges are perpendicular to the bases, then the
lateral faces will be rectangles. (Why?) In such a case, we put the
word right in front of the name of the prism. In this book, the word
box will often be used to refer to a right prism.
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5*"^^!
1
i
1
Right Triangular Prism
Right Pentagonal Prism
Note The base ofa right triangular prism is not necessarily a right
triangle.
Part Two: Sample Problem
Problem
Given: The right triangular prism
shown
Find: a Itslateral area [LA.)
b Its total area [T.A.]
Solution
The right triangular prism can be divided into two triangles (the
parallel bases) and three rectangles (the lateral faces).
L.A. =
A = 14 • 20
A =13*20
= 260
13
20
= 280
A=15«20
14
20
20
Thus, L.A. = 260 + 280 + 300 = 840.
b T.A. = L.A.
14
14
Base
Base
Since the area of each base is |(12)(14), or 84,
T.A. = 840 + 84 + 84 = 1008.
562
Chapter 12 Surface Area and Volume
= 300
15
Problem Set A
1 Find the total surface area of a right
rectangular prism with the given
dimensions.
a € = 15 cm, w = 5 cm, h = 10 cm
b I = 12 mm, w = 7 mm, h = 3 mm
c I = 18 in., w = 9 in., h = 9 in.
2 Find the lateral area of a right triangular
prism with the given dimensions.
a € = 10, a = 3, b = 5, c = 7
b € = 14, a = 2, b = 3, c = 4
3 A right triangular prism has bases that
are isosceles triangles. What is
a The prism's lateral area?
b The area of one base?
c The prism's total area?
4 Find the total surface area of a right
equilateral triangular prism with the giv
en dimensions.
a s = 6, e = 5
b s = 12, € = 10
A cube is a rectangular prism in which
each face is a square. What is the total
/•
surface area of a cube in which each
y
t
~~s
edge has a measure of
a 5?
Problem Set B
6 Find the total area of the pieces of cardboard needed to con
struct each open box shown.
a
Open Top
b
Section 12.1
Surface Areas of Prisms
563
Problem Set B, continued
7 Find the lateral area and the total area of each prism.
a Right Square Prism
c Right Isosceles Triangular Prism
b Right Triangular Prism
d Regular Hexagonal Prism
6/ \6
6[
/ \
6___ !
6\ A
W
8 Find the total area of the right prism
shown.
Problem Set C
9 Find the lateral area and the total area of
the right prism shown.
10 The perimeter of the scalene base of a pentagonal right prism is
17, and a lateral edge of the prism measures 10. Find the prism's
lateral area.
11 A 6-inch cube is painted on the outside and cut into 27 smaller
cubes,
a How many of the small cubes have six
faces painted? Five faces painted? Four
faces painted? Three faces painted?
Two faces painted? One face painted?
No face painted?
b If one of the small cubes is selected at
random, what is the probability that it
has at least two painted faces?
c What is the total area of the unpainted
surfaces?
Chapter 12 Surface Area and Volume
/
/
/
/
/
/
/
vr
/J
Surface Areas of Pyramids
Objective
After studying this section, you will be able to
0 Find the surface areas of pyramids
Part One: Introduction
Triangular Pyramid
Rectangular Pyramid
Pentagonal Pyramid
A pyramid has only one base. Its lateral edges are not parallel but
meet at a single point called the vertex. The base may be any type of
polygon, but the lateral faces will always be triangles. The diagrams
above show three types of pyramids. Notice that each pyramid is
named by its base.
A regular pyramid has a regular polygon as
its base and also has congruent lateral edges.
Thus, the lateral faces of a regular pyramid
are congruent isosceles triangles.
Recall from Section O.G 'hcL Llic JliUdu o± a regular pyramid is a
perpendicular segment from the vertex to the base. (The foot of the
altitude is the center of the base.) Also recall that a regular pyra
mid's slant height is the height of a lateral face.
Slant
Height
Altitude
Altitude
Lateral
Edge
The altitude and a slant height
determine a right triangle.
The altitude and a lateral edge
determine a right triangle.
Section 12.2
Surface Areas of Pyramids
565
Part Two: Sample Problems
Problem 1
Given: The regular pyramid shown at
the right
Find: a Its lateral area [L.A.)
b Its total area [T.A.)
Solution
a The lateral area is the sum of the areas of four congruent isosceles
triangles.
The Pythagorean Theorem shows the
10
slant height to be 8. The area of each
lateral face is f(12)t8), or 48, so
T
a
_
^ / /_/____\.v \
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b The total area is equal to the lateral area plus the area of the base.
The area of the square base is 122, or 144, so
T.A. = 192 + 144 = 336.
Problem 2
The base of rectangular pyramid
ABCDE is 10 by 18. The altitude is
12. The lateral edges are congruent.
a Why is ABCDE not a regular
pyramid?
b Find the pyramid's total surface
area.
Solution
a The base is not regular, so
ABCDE is not regular.
b AH and AG are the heights of the
lateral faces. Applying the Pytha
gorean Theorem to AAFH and
AAHG, we find that AH = 13 and
AG = 15. There are five faces:
T.A. =|(18)(13) +|(18)(13) +|(10)(15) +|(10)(15) + (18)(10)
=
117
+
= 564
566
Chapter 12 Surface Area and Volume
117
+
75
+
75
+
180
Part Three: Problem Sets
1 The pyramid shown is regular and has a
square base.
a Find the area of each lateral face.
b Find the pyramid's lateral area.
c Find the pyramid's total area.
2 The pyramid shown is regular and has a
triangular base. What is
a The area of each lateral face?
b The area of the base?
c The total area?
3 The pyramid shown has a rectangular
base, and its lateral edges are congruent.
a Why is this pyramid not regular?
b What is its lateral area?
c What is its total area?
4 The diagram shows a solid that is a com
bination of a prism and a regular
pyramid.
/A--\
\
a Is ABCD a face of the solid?
b How many faces does this solid have?
c Find the total area.
fs
A
10
b
B^
iM
10
A,
F
PRXYZ is a regular pyramid. The mid
points of its lateral edges are joined to
form a square. ABCD. PR = 10 and
RX = 12.
a Find the lateral area of PRXYZ.
b Find the lateral area of pyramid
PABCD.
c What is the area of square ABCD?
d What is the area of square RXYZ?
e Find the ratio of the area of ABCD to
the area of RXYZ.
f What is the area of trapezoid ABXR?
Section 12.1 Surface Areas of Pyramids
567
Problem Set B
6 A regular pyramid has a slant height of 8. The area of its square
base is 25. Find its total area.
7 A regular pyramid has a slant height of
12 and a lateral edge of 15. What is
a The perimeter of the base?
b The pyramid's lateral area?
c The area of the base?
d The pyramid's total area?
8 PABCD is a regular square pyramid.
a If each side of the base has a length of
14 and the altitude (PQ) is 24, find the
pyramid's lateral area and total area.
b If each slant height is 17 and the alti
tude is 15, find the pyramid's lateral
area and total area.
9 Suppose that the pyramid in problem 8 were not regular but had
a rectangular base and congruent lateral edges.
a Given that PQ = 8, CD = 12, and BC = 30, find PR (the slant
height of face PCD), PS (the slant height of face PBC), and the
lateral area and the total area of the pyramid.
b If each lateral edge were 25 and the base were 24 by 30, what
would the altitude (PQ) of the pyramid be?
Problem Set C
10 Each lateral edge of a regular square pyramid is 3, and the
height of the pyramid is 1. What is
a The measure of a diagonal of the base?
b The pyramid's slant height?
c The area of the base?
d The pyramid's lateral area?
11 A regular tetrahedron ("four faces") is a pyramid with four equi
lateral triangular faces. If a regular tetrahedron has an edge of 6,
what is
a Its total surface area?
b Its height?
Chapter 12 Surface Area and Volume
12 A regular octahedron is a solid with
eight faces, each of which is an equilat
eral triangle. If each edge of the regular
octahedron shown is 6 mm long, what is
a The solid's total surface area?
b The distance from C to E?
c The distance from A to B?
d The shape of quadrilateral ACBE?
13 A regular hexahedron is a solid that does not have triangular
faces. What is the common name for a regular hexahedron?
C A Ri£•;.£ Rv;P RiO'-..FiI L E
Packaging Ideas
Jolene Randby engineers design
Each year American workers produce more
than $1 trillion worth of manufactured products.
Nearly all must be packaged in some kind of
engineers. Randby also calculates the size of
the pallet, the small platform on which shipping
containers are stacked for storage and transpor
box, carton, bag, or can. Besides basic size con
siderations, economic, environmental, health,
and safety factors also affect the design of a
package. The task of harmonizing all these fac
tation.
tors falls to packaging engineers, employed by
all major manufacturers.
"My job is to write the specifications and
packaging standards
and to design the
packages for all of
the industrial tapes
that we produce,"
explains Jolene
Randby, a packaging
engineer with the
3M Corporation In
St. Paul, Minnesota.
"Our tapes are man
ufactured in roils
varying from 60 yards to 1000 yards in length.
Jolene Randby attended high school in her
hometown of St.' Paul. At the University of Wis
consin at Stout, where she earned her bache
lor's degree, she majored In industrial technolo
gy, with a concentration in packaging
engineering.
High on the list of bene
fits of her job is the
necessary travel. "We have
twelve converting plants
where our packages are
assembled," she exofalns.
"i visit them all to oversee
production." She also
works closely With the mar
keting and purchasing de
partments at 3M. '
Tape measuring 2 in. in
width Is manufactured in 250-foot rolls, each
When a new container is needed, I use basic
measuring 3§ In. In diameter. Ashipping con
geometric formulas to find the roll dimensions
tainer contains four stacks of tape arranged in a
square array, with six roils in each stack-How
and then calculate the size of the primary con
tainer and the shipping container." Formulas for
volume and surface area of rectangular prisms
and cylinders are commonly used by packaging
many shipping cohtajners c^h fit in one layer on
a pallet measuring 42 in. by 48 in.? The walls of
each container are £in. thick.
Section 12.2
Surface Areas of Pyramids
569