introduction to the third...
TRANSCRIPT
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Introduction to the Third Dimension
14.1 Cut, Fold, and Voila!Nets........................................................................... 905
14.2 More Cans in a CubeThe.Cube......................................................................921
14.3 Prisms Can Improve Your Vision!Prisms........................................................................ 941
14.4 Outside and Inside a PrismSurface.Area.and.Volume.of.a.Prism............................953
14.5 The Egyptians Were on to Something—or Was It the Mayans?Pyramids.....................................................................963
14.6 And The Winning Prototype Is . . . ?Identifying.Geometric.Solids..
in.Everyday.Occurrences..............................................979
Rice is central to
the daily diet of billions of people around the world. Ornamental
containers, such as the ceramic canister shown, are common in kitchens where rice
is cooked.
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14.1. . . Nets. . . •. . . 905
Have.you.ever.heard.of.the.term.rebranding?.Generally,.this.term.means.to.
give.a.product.or.an.item.a.new.look..Rebranding.isn’t.a.decision.that.businesses.
take.lightly..Many.times,.marketing.research.is.performed.on.a.product’s.current.
look.and.possible.new.looks..There.is.also.the.risk.that.people.will.not.recognize.
the.product,.perhaps.leading.to.fewer.sales..What.items.or.products.have.you.
seen.that.have.gone.through.rebranding?.Do.you.think.rebranding.only.deals.with.
products.or.items?
Cut, Fold, and Voila!Nets
Key Terms. geometric.solids
. prototype
. edge
. face
. vertex
. net
Learning GoalsIn this lesson, you will:
. Sketch.various.views.of.a.solid.figure.to.provide.a.
two-dimensional.representation.of.a.three-dimensional.figure.
. Construct.a.net.from.a.model.of.a.geometric.solid.
. Construct.a.model.of.a.geometric.solid.from.a.net.
. Use.nets.to.provide.two-dimensional.representations.of.a.
geometric.solid.
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Problem 1 Prototype #1
Geometric solids are all bounded three-dimensional geometric figures. The three
dimensions are length, width, and height.
The new marketing director of a rice distribution company, Rice Is Nice, has decided
to change the way its product is packaged. The marketing director hopes to get more
people to notice and talk about the product. She assigned her product development team
to create prototypes. A prototype is a working model of a possible new product. Each
prototype needs to be a different-shaped container to package the product.
Rice is Nice is considering changing the dimensions of its current packaging. The box
shown is one prototype.
Prototype#1
The height of the box is 5.7 centimeters, the width or depth of the box is 2.9 centimeters,
and the length of the box is 4.3 centimeters.
1. Use the figure shown to answer each question.
a. How many sides of the box can you see?
b. Describe the location of the sides you can see.
c. How many sides can you not see?
d. What sides can you not see?
e. What is the shape of each side?
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14.1. . . Nets. . . •. . . 907
2. Sketch each side of the box, label the location of the side, and include the measurements.
Imagine cutting out each side you sketched and taping
the corresponding edges together to construct the box of
Prototype #1. An edge is the intersection of two faces of
a three-dimensional figure. A face is one of the polygons
that makes up a polyhedron. The point where edges meet is
known as a vertex of a three-dimensional figure.
A net is a two-dimensional representation of a three-dimensional
geometric figure. A net is cut out, folded, and taped to create a model of
a geometric solid.
A vertex of a solid is similar to the vertex of
an angle.
A net has all these properties:
● The net is cut out as a single piece.
● All of the sides of the geometric solid are
represented in the net.
● The sides of the geometric solid are drawn
such that they share common edges.
● The common edges are labeled
as fold lines.
● Tabs are drawn on the edges to
be taped.
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14.1. . . Nets. . . •. . . 909
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3. Cut, fold, and tape this net to create Prototype #1.
Prototype #1
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14.1. . . Nets. . . •. . . 911
Problem 2 Prototype #2
Sandy created this net to model her prototype for rice packaging. Cut out, fold, and tape
this net to create a prototype.
Prototype #2
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14.1. . . Nets. . . •. . . 913
Problem 3 Prototype #3
Emilia created this net to model her prototype for rice packaging. Cut out, fold, and tape
this net to create a prototype.
Prototype #3
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14.1. . . Nets. . . •. . . 915
Problem 4 Prototype #4
Trang created this net to model his prototype for rice packaging. Cut out, fold, and tape
this net to create a prototype.
Pro
toty
pe
#4
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Hang on to your prototype
models, you will use these again in other lessons in
this chapter.
14.1. . . Nets. . . •. . . 917
Problem 5 Using Your Sorting Hat!
The marketing director requires the team members to present their prototypes at the next
Rice Is Nice stockholders’ meeting. She told the team that 4 prototypes are too many.
The team members could not decide which of the prototypes to exclude, so they intend to
group the 4 prototypes into 2 categories and highlight each category.
1. Using all 4 solids, sort them into 2 groups, and explain your reasoning.
2. Compare your method of grouping with your classmates. Did everyone use the same
groupings? Explain your reasoning.
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The net for each prototype you created is shown. You will use these representations again
in this chapter.
Pro
toty
pe
#1
5.7 cm
2.9 cm
4.3
cm
Prototype #2
4.3 cm
4.3 cm
5.7
cm
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14.1. . . Nets. . . •. . . 919
Prototype #3
3.8 cm
5.3 cm
Prototype #4
3.3
cm
5.1 cm
3.7 cm
Be prepared to share your solutions and methods.
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920. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
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14.2. . . The.Cube. . . •. . . 921
Only.90.years.ago,.the.standard.beverage.size.bottle.was.6.5.ounces..Now,.
some.convenient.stores.tout.128-ounce.drinks!.But.the.amount.in.one.serving.
isn’t.the.only.thing.that.has.gotten.bigger..Why.do.you.think.drink.portions.have.
become.larger.over.the.years?.Do.you.think.the.common.practice.of.restaurants.
refilling.drinks.is.a.contributing.factor?
More Cans in a CubeThe Cube
Learning GoalsIn this lesson, you will:
. Create.a.model.of.a.cube.from.a.net.
. Construct.a.model.of.a.geometric.solid.from.a.net.
. Use.nets.to.provide.two-dimensional.representations.of.
a.cube.
. Estimate.the.volume.and.surface.area.of.a.cube.
. Use.nets.to.compute.the.volume.and.surface.area.of.a.
cube.
. Use.a.formula.to.determine.the.volume.of.a.cube.
. Use.unit.cubes.to.estimate.the.surface.area.and.volume.
of.larger.cubes.
. Use.appropriate.units.of.measure.when.computing.the.
surface.area.and.volume.of.a.cube.
. Explore.how.doubling.the.dimensions.of.a.cube.affects.
the.volume.of.the.cube.
Key Terms. point
. line.segment
. polygon
. polyhedron
. regular.polyhedron
. congruent
. cube
. unit.cube
. diameter
. surface.area
. volume
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Problem 1 Speaking a Common Language
Before beginning the lesson, everyone must speak a common language. It is important
to use the same words when studying mathematics and describing geometric terms.
For example, a word you may have used in the past may actually have a more precise
definition when dealing with mathematics. For example, the word point has many
meanings outside of math. However, the mathematical definition of point is a location
in space. A point has no size or shape, but it is often represented by using a dot and is
named by a capital letter. A line segment is a portion of a line that includes two points
and all the points between those two points.
Recall, a polygon is a closed figure formed by three or more line segments. Knowing
these definitions will help you learn the meanings of other geometric words.
1. What do you think is the meaning of a closed figure?
2. Sketch what you think is an example of a polygon.
3. Is your sketch a closed figure? Are all of the sides in your sketch formed by line segments?
4. Compare your sketch with your classmates’ sketches. Did everyone sketch the same
polygon? Explain how your classmates’ and your sketches are the same or different.
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14.2. . . The.Cube. . . •. . . 923
A polyhedron is a three-dimensional figure that has polygons
as faces.
5. Sketch what you think is an example of a polyhedron.
6. Does your sketch look like a three-dimensional figure?
Does your sketch show polygons for every face?
7. Compare your sketch with your classmates’ sketches. Did everyone sketch the same
polyhedron? Explain how the sketches are the same or different.
Would any of the prototypes
you created in the last lesson be
polyhedrons?
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A regular polyhedron is a three-dimensional solid that has congruent regular polygons
as faces and has congruent angles between all faces. Congruent means having the same
size, shape, and measure.
8. Sketch what you think is an example of a regular polyhedron.
9. Does your sketch look like a three-dimensional solid that has congruent regular
polygons as faces and congruent angles between all the faces?
10. Compare your sketch with your classmates’ sketches. Did everyone sketch the same
regular polyhedron? Explain how the sketches are the same or different.
A cube is a regular polyhedron whose six faces are congruent squares.
A unit cube is a cube that is one unit in length, one unit in width, and one unit in height.
In this chapter, unit cubes are used as manipulatives to explore characteristics of
geometric solids. The unit cubes are typically 1 centimeter in length, 1 centimeter in width,
and 1 centimeter in height. For this reason, use a centimeter ruler to measure lengths in
this chapter.
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14.2. . . The.Cube. . . •. . . 925
Problem 2 Is It Really a Cube?
In 1993, beverage manufacturers decided to repackage their product to boost sales.
Research indicated that consumers would rather buy more cans of their favorite
beverages at one time than make several trips to the store. The marketing team came up
with the idea of packaging several cans of their beverage together in a way that was easy
to carry. This packaging is called the “cube.”
A cube contains 24 cans.
1. Sketch some of the possible rectangular arrangements of 24 cans. Your arrangements
may have more than one layer.
The diameter of each can is 2 inches. The diameter is the distance across a circle through
its center. The height of each can is 6 inches.
2. What are the approximate dimensions of rectangular boxes needed to contain each
arrangement of cans you sketched in Question 1?
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3. The manufacturer decided to go with a two layer arrangement and called it a cube.
a. What are the dimensions of this arrangement?
b. Why do you think they made the decision to call this a cube?
c. Explain why calling the package a cube can be confusing.
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Problem 3 Characteristics of a Cube
The cube is a basic geometric solid.
1. Sketch a cube.
2. How many faces of the cube can you see?
3. Describe the location of the faces you can see.
4. How many faces can you not see?
5. Describe the location of the faces that you cannot see.
6. What is known about the length, height, and width of the cube?
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7. Would you measure the length, width, and height of a cube using linear units such as
inches, centimeters, and feet? Or, would you use square units such as square inches,
square centimeters, and square feet? Or, would you use cubic units such as cubic
inches, cubic centimeters, and cubic feet?
8. Sketch and describe the shape of each face of a cube.
9. Is a cube a polygon? Explain your reasoning.
10. Is a cube a polyhedron? Explain your reasoning.
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14.2. . . The.Cube. . . •. . . 929
Problem 4 Cube Net
There are 11 different nets that can be created to model a cube.
1. Here is one example of a net of a cube.
Cut it out, fold it, and tape it together to create a geometric model of a cube.
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14.2 The Cube • 931
2. Describe the number of faces, vertices, and edges of the cube.
3. Not all of these nets create a cube. Circle each figure that is a net of a cube.
4. How did you determine which nets were cubes in Question 3?
Remember, a cube has six faces_and only
six faces!
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Problem 5 Surface Area of a Cube
Surface area is the total area of the two-dimensional surfaces (faces and bases) that
make up a three-dimensional object.
Consider the model of the cube you created in Problem 4, Question 1, to answer
each question.
1. What is true about the area of each of the 6 faces of a cube?
2. Is the area of a face of a cube measured using linear units such as inches,
centimeters, and feet? Or, using square units such as square inches, square
centimeters, and square feet? Or, using cubic units such as cubic inches, cubic
centimeters, and cubic feet?
3. Describe a strategy that you can use to determine the total surface area of a cube?
4. Use a centimeter ruler to calculate the total surface area of your cube.
5. How is the net of a cube helpful when determining the surface area of a cube?
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14.2. . . The.Cube. . . •. . . 933
6. What is the surface area of a unit cube?
7. How are unit cubes helpful when determining the
surface area of a larger cube?
Problem 6 Volume of a Cube
Volume is the amount of space occupied by an object.
The volume of a cube is calculated by multiplying the length times the width times the
height of the cube.
Use the model of the cube you created in Problem 4, Question 1, to answer each question.
1. Would you measure the volume of a cube using linear units such as inches,
centimeters, and feet? Or, would you use using square units such as square inches,
square centimeters, and square feet? Or, would you use using cubic units such as
cubic inches, cubic centimeters, and cubic feet?
2. How is estimating the volume of a cube different from calculating the volume of a cube?
Do you have unit cubes
ready?
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3. Estimate the volume of your cube by stacking unit
cubes next to the model of the cube.
4. Measure the length, width, and height of the cube using a
centimeter ruler. Then, multiply the length, width, and height to
calculate the volume.
5. What is the difference between the estimation of the volume and
the calculation of the volume?
6. What is the ratio of the difference between the estimation and the
calculation of the volume to the calculation of the volume?
7. Write the ratio from Question 6 as a percent. This is the percent of increase or
decrease in volume resulting from estimation.
8. How is the net of a cube helpful when determining the volume of a cube?
9. What is the volume of a unit cube?
10. How could unit cubes be helpful when you are determining the volume of a larger cube?
“Grab a handful of unit cubes!
What percent difference is considered a good
estimate?
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14.2. . . The.Cube. . . •. . . 935
Problem 7 Volume Formula
Volume can be determined by using the formula V 5 3 w 3 h, where V is the volume
of the cube, is the length of the cube, w is the width of the cube, and h is the height of
the cube.
The base of a cube is a square. Recall that the area of a square is calculated by
multiplying the length of the square by the width of the square. Written as a formula, the
area of the base of a cube is Area of the Base 5 3 w.
Consider the two formulas:
V 5 3 w 3 h
Area of the Base 5 3 w
If B is used to represent the area of the base of a cube, then
you can rewrite the second formula as: B 5 3 w.
Now consider the two formulas:
V 5 3 w 3 h
B 5 3 w
Using both of these formulas, you can rewrite the formula for
the volume of a cube as V 5 B 3 h, where V represents the
volume of the cube, B represents the area of the base of the
cube, and h represents the height of the cube. You can use this
formula to calculate the volume of many different geometric
solids. However, the formula for calculating the value of B will
change depending on the base shape of the polyhedron.
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Use the formula V 5 B 3 h to answer each question.
1. The length, width, and height of the cube are each equal to 2 centimeters.
2 cm
2 cm
2 cm
a. Calculate the area of the base of the cube.
b. What is the height of the cube?
c. Calculate the volume of the cube.
2. The volume of this cube is 27 cubic centimeters.
27 cm3
a. What is the area of the base of the cube?
b. What is the height of the cube?
“Keep in mind that a number doesn't say much without
a label.
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14.2. . . The.Cube. . . •. . . 937
Problem 8 Jerome and Roberta Need Your Help!
Jerome began stacking unit cubes to make a larger cube but was interrupted before he
could finish. The figure shown displays how much progress Jerome made in making the
larger cube.
1. You can see how long, wide, and tall Jerome wanted the cube. Calculate the volume
and surface area of Jerome’s cube if he had completed it.
Roberta is using unit cubes to determine the surface area and volume of larger cubes. She
wants to build 6 different size cubes to compare the surface area and volume, but she
realized that she would not have enough unit cubes to complete the models. She decides
to just build the length, width, and height of the first four cubes and to look for a pattern.
The figures Roberta built are shown.
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2. Roberta thinks she sees a pattern, but she needs to sketch the fifth and sixth cube to be
sure. Help Roberta by sketching the fifth and six figures based on the pattern you see.
3. Roberta is organizing the data in a table. Help Roberta complete the table.
Dimensions of the Cube
Area of One Side of the Cube
(in square units)
Surface Area of the Cube
(in square units)
Volume of the Cube
(in cubic units)
1.3.1.3.1
2.3.2.3.2
3.3.3.3.3
4.3.4.3.4
5.3.5.3.5
6.3.6.3.6
4. Describe how Roberta can use the dimension (length or width) of a cube to determine
the area of one side of the cube.
5. Describe how Roberta can use the area of one side of the cube to determine the
surface area of the cube.
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6. Describe how Roberta can use the dimensions (length, width, and height) of the cube
to determine the volume of the cube.
7. Roberta is looking at the completed table and notices that when the dimensions of a
cube are doubled, the volume of the larger cube is predictable. She saw the pattern!
Describe the pattern Roberta sees in the completed table.
8. Use Roberta’s pattern and the volume of a 5 3 5 3 5-unit cube to predict the volume
of a 10 3 10 3 10-unit cube.
9. Use Roberta’s pattern and the volume of a 10 3 10 3 10-unit cube to predict the
volume of a 20 3 20 3 20-unit cube.
10. Roberta’s lab partner, Derrick, looked at her completed table in Question 3 and found it
interesting that the surface area and the volume of a 6 3 6 3 6-unit cube is 216. Roberta
helped Derrick understand that they were not equal. What did Roberta tell Derrick?
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940. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
Talk the Talk
Each numerical answer describes the volume or the surface area of a cube. Which is it?
How do you know?
1. 125
5
2. 24
2
3. 13.5 m2
4. 3.375 m3
Be prepared to share your solutions and methods.
“Labels really do matter.
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14.3. . . Prisms. . . •. . . 941
Key Termsprism
bases.of.a.prism
lateral.faces
height.of.a.prism
rectangular.prism
right.prism
Learning GoalsIn this lesson, you will:
. Sketch.a.model.of.a.right.rectangular.prism.
. Create.models.of.various.prisms.
. Determine.the.characteristics.of.various.prisms.
Do.you.know.that.binoculars.and.prisms.are.close.friends?.In.1854,.Ignazio.
Porro.realized.this.and.patented.the.“Porro.Prism.”.Using.right.triangular.prisms.
he.was.able.to.turn.an.image.right-side.up..Basically,.when.an.image.is.gathered.
by.the.lens,.the.image.is.upside.down..Thus,.inside.each.eyepiece.are.two.prisms,.
which.turn.the.image.right-side.up.and.invert.the.image.from.left.to.right..What.
do.you.think.would.happen.if.only.one.prism.was.used.in.binoculars?.What.do.you.
think.would.happen.if.4.prisms.were.used.in.binoculars?
Prisms Can Improve Your Vision!Prisms
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942. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
Problem 1 Getting to Know Prisms!
A prism is a polyhedron with two parallel and congruent faces and all the other faces
are parallelograms. These two parallel and congruent faces are known as the bases of a
prism. The remaining parallelogram-shaped faces are known as lateral faces.
1. What do you think “two parallel and congruent faces” means?
2. What is a parallelogram?
3. Sketch what you think is an example of a prism.
4. Does your sketch show two bases that are the same polygon and
are they drawn parallel to each other? Are all of the faces of your
polyhedron parallelograms except for the bases?
5. Identify the bases in your sketch.
Look at the prototype models
you created earlier for ideas.
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14.3. . . Prisms. . . •. . . 943
6. Compare your sketch with your classmates’ sketches. Did everyone sketch the same
prism? Explain how the sketches are the same or different.
A height of a prism is the length of a line segment that is drawn from one base to the
other base. This line segment must be perpendicular to the other base.
7. Use your sketch to explain what is meant by “height of a prism”?
A rectangular prism is a prism that has a rectangle as its base.
8. Sketch what you think is an example of a rectangular prism.
9. Are all of the faces of your sketch rectangles? Does your sketch have two bases that
are parallel and congruent?
10. Compare your sketch with your classmates’ sketches. Did everyone sketch the same
rectangular prism? Explain how the sketches are the same or different.
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944. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
A right prism is a prism that has bases aligned one directly above the other and has
lateral faces that are rectangles. All prisms associated with this chapter are right prisms.
11. Which faces of a prism are considered “lateral faces”?
12. Sketch what you think is an example of a right prism.
13. Are the bases of your prism aligned one directly above the other? Are all of the lateral
faces in your sketch rectangles?
14. Compare your sketch with your classmates’ sketches. Did everyone sketch the same
right prism? Explain how the sketches are the same or different.
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14.3. . . Prisms. . . •. . . 945
Problem 2 Right Rectangular Prism
1. Use the model of the cube you created in Lesson 14.2 to answer each question.
a. Is a cube a prism? Explain your reasoning.
b. Is a cube a rectangular prism? Explain
your reasoning.
c. Is a cube a right prism? Explain your reasoning.
2. In your own words, describe what makes up a right rectangular
prism.
Can a square be a rectangle? Can a rectangle be a
square? Ah! My head is spinning!
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946. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
3. Sketch a right rectangular prism that is not a cube.
a. How many faces can you see?
b. Describe the location of the faces you can see.
c. How many faces can you not see?
d. Describe the location of the faces you cannot see.
e. What is known about the length, width, and height of the right rectangular prism?
f. Describe the shape of each face of the right rectangular prism.
“Have you created any other
model that is a right rectangular prism?
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14.3 Prisms • 947
Problem 3 Characteristics of a Prism
Use pasta and miniature marshmallows to construct a
model of each prism shown. Use your model to answer
questions about the prisms.
1. Construct and analyze this prism.
a. Name the polygon that is the base of this prism.
b. How many faces of the prism are lateral faces?
c. Identify the number of vertices, edges, and faces.
d. How is a height of this prism determined?
You can break the pasta
to be any length you want.
948 • Chapter 14 Introduction to the Third Dimension
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2. Construct and analyze this prism.
a. Name the polygon that is the base of this prism.
b. How many faces of the prism are lateral faces?
c. Identify the number of vertices, edges, and faces.
d. How is a height of this prism determined?
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3. Construct and analyze this prism
a. Name the polygon that is the base of this prism.
b. How many faces of the prism are lateral faces?
c. Identify the number of vertices, edges, and faces.
d. How is a height of this prism determined?
14.3 Prisms • 949
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950 • Chapter 14 Introduction to the Third Dimension
4. Construct and analyze this prism.
a. Name the polygon that is the base of this prism.
b. How many faces of the prism are lateral faces?
c. Identify the number of vertices, edges, and faces.
d. How is a height of this prism determined?
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14.3. . . Prisms. . . •. . . 951
5. Complete the table shown with the data from Questions 1, 2, 3, and 4.
Shape of the Base of Prism
(Regular Polygon)
Number of Sides
of the Base
Number of Vertices
Number of Edges
Number of Faces
6. Use the data from the table in Question 5 and any patterns you notice to answer
each question.
a. What is the relationship between the number of sides of the base and the number
of vertices of each prism?
b. What is the relationship between the number of sides of the base and the number
of edges of each prism?
c. What is the relationship between the number of sides of the base and the number
of faces of each prism?
7. Without making a model or drawing a sketch, predict the number of vertices, edges,
and faces for an octagonal prism. Describe your reasoning for making the prediction.
8. To verify your prediction, make a model of an octagonal prism to check your answers.
Be prepared to share your solutions and methods.
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952. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
14.4. . . Surface.Area.and.Volume.of.a.Prism. . . •. . . 953
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Many.zoos.and.aquariums.that.hold.large.marine.animals.must.order.or.build.
custom-made.tanks.for.their.creatures..These.tanks.come.in.many.different.
shapes.and.sizes..A.large.fish.tank.that.might.hold.jellyfish.or.sea.horses.at.the.
zoo.might.have.dimensions.of.48"..24"..25".and.could.hold.around.115.gallons.
of.water..Tanks.for.large.animals.such.as.dolphins.and.whales.could.have.
dimensions.of.46'..23'..30'..That.means.these.tanks.hold.around.238,000.
gallons.of.water!.How.do.you.think.the.zoo.keepers.determine.what.size.the.tank.
should.be.and.how.much.water.it.should.hold?
Outside and Inside a PrismSurface Area and Volume of a Prism
Learning GoalsIn this lesson, you will:
. Use.unit.cubes.to.estimate.the.volume.and.surface.area.of.a.right.rectangular.prism.
. Use.nets.to.compute.the.volume.and.surface.area.of.a.right.rectangular.prism.
. Use.a.formula.to.determine.the.volume.of.a.right.rectangular.prism.
. Use.appropriate.units.of.measure.when.computing.the.surface.area.and.volume.of.a.
right.rectangular.prism.
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954. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
Problem 1 Right Rectangular Prism Net
1. The net shown is Prototype 1 from the first lesson.
5.7 cm
2.9 cm4.
3 cm Prototype
#1
a. Write the name of each side on each face: front, back, top,
bottom, left side, and right side.
b. Use the net to estimate the surface area of the right rectangular
prism. Recall that the unit of measurement when calculating the
surface area is square units.
c. Calculate the surface area of the right rectangular prism.
Explain your calculation.
“Get out your prototype 1 model to help answer
questions in this lesson.
Pairs of faces in the prism
are congruent. This can help me
estimate.
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14.4. . . Surface.Area.and.Volume.of.a.Prism. . . •. . . 955
d. How does the estimation of the surface area compare to the calculation of the
surface area?
e. Use your model of a right rectangular prism to determine the number of faces,
vertices, and edges.
f. Estimate the maximum number of unit cubes that would fit inside your model of a
right rectangular prism.
Calculating the actual volume of a right rectangular prism is similar to calculating the
actual volume of a cube. Multiply the length of the rectangular prism times the width of
the rectangular prism times the height of the rectangular prism, or calculate the area of the
base and multiply the product by the height.
g. Calculate the volume of the right rectangular prism. Recall that the unit of
measurement when calculating the volume is cubic units.
h. How does the estimation of the volume compare to the calculation of the volume?
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956. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
2. Place your model of the right rectangular prism you created on your desk such that it
rests on one of the largest sides.
a. Lightly sketch a letter X on the two bases of the prism.
b. Now, turn your model of the right rectangular prism such that it rests on one of the
smallest sides and lightly sketch a letter X on the two bases of the prism.
c. Are there two Xs on any face of the prism?
d. Think of the front, back, left, right, bottom, and top faces as locations on the prism.
Is the location of the bases in part (a) the same location as the bases in part (b)?
e. In your own words, describe how you can determine which sides of a right
rectangular prism are bases.
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14.4. . . Surface.Area.and.Volume.of.a.Prism. . . •. . . 957
Problem 2 Surface Area of a Prism
The surface area of the cube was calculated by adding all of the areas of all of the faces of
the cube. The same process is also used to determine the surface areas of prisms.
Similar to the cube, the area of a base of a rectangular prism can be calculated using the
area formula, B 5 3 w, where B is the area of the base, is the length of the rectangular
base, and w is the width of the rectangular base.
However, many of the prisms in this lesson do not have rectangular bases. The base of
a prism can be a variety of polygons. If the base of the prism is a pentagon, a hexagon,
an octagon, or any polygon different from a rectangle, you need to use a strategy to
determine the area of the base.
One base of a regular pentagonal prism is shown. Recall that when a polygon is regular,
that means all of the sides of the polygon are equal in length and all of the angles of the
polygon are equal in measure.
1. Locate and place a point at the center of the pentagon. From the center point, draw
line segments to connect the point with each vertex of the pentagon.
2. Describe the new polygons formed by adding these line segments.
3. What information do you need to calculate the area of each new polygon?
4. What formula is used to calculate the area of each new polygon?
5. Describe a strategy to determine the area of the entire pentagonal base.
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958. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
One base of a regular hexagonal prism is shown.
6. Use the same strategy you used in Question 1 to divide the hexagon into new polygons.
7. Describe the new polygons formed by adding these line segments.
8. What information do you need to calculate the area of each new polygon?
9. What formula is used to calculate the area of each new polygon?
10. Describe a strategy to determine the area of the entire hexagonal base.
11. Do you think this strategy works for any regular polygonal base of a prism? Explain
your reasoning.
In conclusion, the surface area of a prism is the sum of the areas of all of the lateral faces
of the prism plus the areas of the two bases.
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14.4. . . Surface.Area.and.Volume.of.a.Prism. . . •. . . 959
Problem 3 Volume Formula of a Prism
The formula for calculating the volume of a cube is V 5 B 3 h, where V represents the
volume of the cube, B is the area of the base of the cube, and h is the height of the cube.
The same is true for prisms.
Use the same formula, but apply it to a prism.
1. What does the variable V represent?
2. What does the variable B represent?
3. What does the variable h represent?
4. Write the formula for determining the volume of a prism. Define all variables used in
the formula.
5. Describe the strategy used for determining the area of the base of the prism when the
base is a regular polygon but not rectangular.
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960. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
Talk the Talk
Each numerical answer describes the volume or the surface area of a right rectangular
prism. Which is it? How do you know?
1. 13.44
2 cm
4.8 cm
1.4 cm
2. 33.64
1.6 cm
5.2 cm 1.25 cm
3. 42.5 m2
4. 50.8 m3
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14.4. . . Surface.Area.and.Volume.of.a.Prism. . . •. . . 961
Prisms are named by the shape of their bases.
5. Name the polygons that best describe the bases of each prism.
a. a pentagonal prism
b. an octagonal prism
c. a triangular prism
d. a decagonal prism
e. a hexagonal prism
f. a heptagonal prism
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962. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
6. Use the nets shown to determine the name of each prism.
a. b.
c. d.
e. f.
Be prepared to share your solutions and methods.
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14.5. . . Pyramids. . . •. . . 963
The Egyptians Were on to Something—or Was It the Mayans? Pyramids
The.Great.Pyramids.of.Egypt.are.a.favorite.tourist.spot.for.any.travelers.in.
the.area..Not.to.be.outdone,.the.pyramids.of.Mexico.are.also.a.favorite.tourist.
attraction.and.quite.challenging.to.climb!.Egypt.does.have.what.is.considered.to.
be.the.oldest.pyramid.in.the.world..The.Step.Pyramid.in.Saqqara.is.considered.to.
be.the.oldest.stone.pyramid..Some.experts.date.the.pyramid.was.built.between.
2649.and.2575.BC!.How.do.you.think.archaeologists.determine.the.age.of.a.
structure?.Do.you.think.there.was.a.reason.why.two.civilizations.were.alike.in.
building.pyramids?
Key Terms. pyramid
. vertex.of.a.pyramid
. height.of.a.pyramid
. slant.height.of.a.pyramid
Learning GoalsIn this lesson, you will:
. Create.a.model.of.a.pyramid.from.a.net.
. Use.nets.to.provide.two-dimensional.representations.
of.a.pyramid.
. Use.nets.to.estimate.the.surface.area.of.a.pyramid.
. Use.appropriate.units.of.measure.when.computing.
the.surface.area.of.a.pyramid.
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964. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
Problem 1 Getting to Know Pyramids
A pyramid is a polyhedron with one base and the same number of triangular faces as
there are sides of the base. The triangular faces are called lateral faces.
The vertex of a pyramid is the point at which all lateral
faces intersect.
All of the pyramids associated with this chapter have a
vertex that is located directly above the center point of
the base of the pyramid.
1. Sketch the first thing that comes to your mind when you
hear the word pyramid.
2. Does your sketch have one base? Does your sketch have the
same number of triangular faces as there are sides of the base?
3. Identify the vertex of the pyramid on your sketch.
4. Compare your sketch with your classmates’ sketches. Did everyone sketch the same
pyramid? Explain how the sketches are the same or different.
Don't pyramids have other
vertices also?
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14.5. . . Pyramids. . . •. . . 965
Similar to a triangle, a height of a pyramid is the length of a line segment drawn from the
vertex of the pyramid to the base. This line segment is perpendicular to the base.
5. Use your sketch to explain what is meant by the “height of a pyramid.”
A slant height of a pyramid is the distance measured along
a lateral face from the base to the vertex of the pyramid
along the center of the face. As shown, a slant height, s, is
the altitude of a triangular lateral face of the pyramid.
S
The height of a triangular face of a pyramid is a dimension often needed to
calculate the total surface area of the pyramid.
Do you notice the
right angle symbol where the slant
height touches the base?
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966. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
Problem 2 Characteristics of a Pyramid
Use pasta and miniature marshmallows to construct a model of each pyramid. Use your
model to answer questions about the pyramids.
1. Construct and analyze this pyramid.
a. Name the polygon that is the base of this pyramid.
b. How many faces of the pyramid are lateral faces?
c. Describe the intersection of all of the lateral faces.
d. How many vertices, edges, and faces are in your model?
e. How can you determine the height of your pyramid?
Just like last time, you can break the pasta to any length
you want.
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14.5. . . Pyramids. . . •. . . 967
2. Construct and analyze this pyramid.
a. Name the polygon that is the base of this pyramid.
b. How many faces of the pyramid are lateral faces?
c. Describe the intersection of all of the lateral faces.
d. How many vertices, edges, and faces are there?
e. How can you determine the height of your pyramid?
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968. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
3. Construct and analyze this pyramid.
a. Name the polygon that is the base of this pyramid.
b. How many faces of the pyramid are lateral faces?
c. Describe the intersection of all of the lateral faces.
d. How many vertices, edges, and faces are there?
e. How can you determine the height of this pyramid?
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14.5. . . Pyramids. . . •. . . 969
4. Construct and analyze this pyramid.
a. Name the polygon that is the base of this pyramid.
b. How many faces of the pyramid are lateral faces?
c. Describe the intersection of all of the lateral faces.
d. How many vertices, edges, and faces are there?
e. How can you determine the height of your pyramid?
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970. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
5. Organize the data from Questions 1, 2, 3, and 4 by completing the table shown.
Shape of the Base of Pyramid
(Regular Polygon)
Number of Sidesof the Base
Number of Vertices
Number of Edges
Number of Faces
6. Use the table you completed in Question 5 to answer each question.
a. What is the relationship between the number of sides of the base and the number
of vertices of each pyramid?
b. What is the relationship between the number of sides of the base and the number
of edges of each pyramid?
c. What is the relationship between the number of sides of the base and the number
of faces of each pyramid?
7. Without making a model or drawing a sketch, predict the number of vertices, edges, and
faces for an octagonal pyramid. Describe your reasoning for making your prediction.
8. To verify your prediction, make a model of an octagonal pyramid to check your answers.
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14.5. . . Pyramids. . . •. . . 971
Problem 3 Pyramid Net
Mr. Morris instructed his math students to use their straw-and-marshmallow models of a
square pyramid to help them create a net.
Shawna raised her hand and said that she had an idea. She said that all she had to do
was remove the marshmallow that was at the top of the pyramid, lower the straws that
formed the lateral sides, and reuse the marshmallow somewhere else, but she would need
3 additional marshmallows to complete the net.
1. Sketch Shawna’s net of a square pyramid. Explain why she would need
3 additional marshmallows.
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972. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
2. Mr. Morris instructed his students to create a model of a pentagonal pyramid.
Then, he wanted them to create a net from their model. If Shawna uses the same
strategy she used to create the net for the square pyramid, how many additional
marshmallows will she need to build a net for a regular pentagonal pyramid?
3. Create Shawna’s net for a regular pentagonal pyramid.
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14.5 Pyramids • 973
4. Allen raised his hand and claimed that he created a different regular pentagonal
pyramid net. A drawing of Allen’s net is shown. How many additional marshmallows
will Allen need?
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974. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
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14.5. . . Pyramids. . . •. . . 975
Problem 4 Surface Area of a Pyramid
Use Allen’s net of a regular pentagonal pyramid to estimate the surface area of a pyramid.
1. What information would you need to estimate the area of one triangle in Allen’s net?
2. Describe a strategy to estimate the area of the base of the pyramid in Allen’s net.
3. Use Allen’s net and a centimeter ruler to estimate the surface area of the pyramid.
Recall that the unit of measurement when estimating surface area is square units.
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4. Shawna is not convinced that Allen’s net is a pyramid because it looks different than
hers. Help to convince Shawna by cutting out, folding, and taping Allen’s net to show
it forms a regular pentagonal pyramid.
5. Do you think the strategy used to calculate the surface area of the regular pentagonal
pyramid also work for pyramids that have different regular polygonal bases?
You have just used various strategies to calculate surface area. However at this point, you
will not generate a formula for determining the surface area of a pyramid. You will explore
the surface area of a pyramid in depth when you study geometry in high school.
976. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
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Talk the Talk
1. Solve for the surface area of each pyramid.
a. A square pyramid where the length of each side of the base is 10 inches and the
slant height is also 10 inches.
b. The base of the pyramid is a regular pentagon.
6 cm
14 cm
3.5 cm
Like prisms, pyramids are named by the shape of their bases.
2. Name the polygon that best describes the base of each pyramid.
a. a pentagonal pyramid b. an octagonal pyramid
c. a triangular pyramid d. a decagonal pyramid
e. a hexagonal pyramid f. a heptagonal pyramid
14.5. . . Pyramids. . . •. . . 977
978. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
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3. Use the nets to determine the name of each pyramid.
a. b.
c. d.
e. f.
Be prepared to share your solutions and methods.
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Learning GoalsIn this lesson, you will:
. Identify.geometric.solids.
. Compare.and.contrast.the.surface.area.of.geometric.solids.
. Apply.the.surface.area.concept.to.a.real-world.situation.
And The Winning Prototype Is . . . ?Identifying Geometric Solids in Everyday Occurrences
When.was.the.last.time.you.saw.a.circle?.Or.perhaps,.when.was.the.last.time.
you.saw.a.line—in.the.geometric.terms?.In.fact,.when.you.begin.to.formally.study.
geometry.in.high.school,.most.of.your.instruction.will.begin.with.two-dimensional.
figures;.however,.the.world.is.full.of.three-dimensional.objects..Even.a.piece.of.
paper.may.“appear”.to.be.a.two-dimensional.object,.but.it.isn’t!.It.does.have.a.
depth,.even.though.that.depth.is.quite.small..
Why.do.you.think.that.most.geometry.courses.start.with.two-dimensional.
examples?.Do.you.think.there.are.some.principles.that.are.key.in.two-dimensional.
examples.that.will.be.used.when.studying.three-dimensional.objects?
14.6. . . Identifying.Geometric.Solids.in.Everyday.Occurrences. . . •. . . 979
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980. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
Problem 1 Geometric Solids are Everywhere!
1. Geometric solids appear in real life in a variety of places. Identify each solid.
a. b.
c. d.
e. f.
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14.6. . . Identifying.Geometric.Solids.in.Everyday.Occurrences. . . •. . . 981
g. h.
i. j.
k. l.
m. n.
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982. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
Problem 2 Gathering Information
Throughout this chapter, you have estimated and calculated the volume and surface area
of four prototypes developed by the Rice Is Nice product development team.
It is now time to compile this information and develop a business plan to market each
prototype. The amount of money it costs the manufacturers to package a product and the
amount of money generated by the sale of this product will determine the profit margin.
1. Complete the table with the information of each prototype for Rice Is Nice.
Prototype Number
Name of the Geometric Solid
Surface Area (in cm2)
Prototype.#1
Prototype.#2
Prototype.#3
Prototype.#4
The nets representing each
of the prototypes can be found at the end of
the first lesson in this chapter.
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14.6. . . Identifying.Geometric.Solids.in.Everyday.Occurrences. . . •. . . 983
2. Match each sketch with the appropriate prototype number.
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984. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
3. Consumers are concerned about how much of the
product they get for their money. If the cost of each
rice container is the same, does the measurement
of the surface area help them to determine the
best buy? Explain.
4. Manufacturers are concerned with maximizing their profit. Does the
measurement of the surface area help them to determine the best
choice? Explain.
Problem 3 Commemorative Canisters
The product development team members came up with a great idea to introduce their new
rice container. They decided to give consumers a complimentary commemorative metal
canister with their first purchase of the newly packaged product. The metal canister will
maintain the same size and same shape as the prototype container.
The stockholders of the Rice Is Nice Company asked the development team to calculate
the cost of materials used to make the commemorative canisters for the four prototypes.
The team wants to compare the price of using aluminum, tin, and copper.
1. Calculate the cost of using aluminum. One rectangular sheet of aluminum measuring
25.4 centimeters long, 10.2 centimeters wide, and 0.04 centimeters thick will
cost $2.69.
a. How many square centimeters are in one rectangular sheet of aluminum?
b. Determine the cost of aluminum per square centimeter to the nearest tenth of
a cent. When calculating an amount of money, always round up.
How are surface area and volume related?
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14.6. . . Identifying.Geometric.Solids.in.Everyday.Occurrences. . . •. . . 985
c. Use the information from Problem 2 to complete the table.
Prototype Number
Name of the Geometric Solid
Surface Area (cm2)
Cost of Aluminum (dollars)
Prototype.#1
Prototype.#2
Prototype.#3
Prototype.#4
2. Calculate the cost of using tin. One rectangular sheet of tin measuring
25.4 centimeters long, 10.2 centimeters wide, and 0.02 centimeters thick will
cost $3.09.
a. How many square centimeters are in one rectangular sheet of tin?
b. Determine the cost of tin per square centimeter to the nearest tenth of a cent.
When calculating an amount of money, always round up.
c. Use the information from Problem 2 to complete the table.
Prototype Number
Name of the Geometric Solid
Surface Area (cm2)
Cost of Tin (dollars)
Prototype.#1
Prototype.#2
Prototype.#3
Prototype.#4
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986. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
3. Calculate the cost of using copper. One rectangular sheet of copper measuring
25.4 cm long, 10.2 cm wide, and 0.06 cm thick will cost $9.49
a. How many square centimeters are in one rectangular sheet of copper?
b. Determine the cost of copper per square centimeter to the nearest tenth of a cent.
When calculating an amount of money, always round up.
c. Use the information from Problem 2 to complete the table.
Prototype Number
Name of the Geometric Solid
Surface Area (cm2)
Cost of Copper (dollars)
Prototype.#1
Prototype.#2
Prototype.#3
Prototype.#4
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14.6. . . Identifying.Geometric.Solids.in.Everyday.Occurrences. . . •. . . 987
Talk the Talk
Team up with a few classmates to write a report to the director of marketing. In the report,
recommend one of the three prototypes for production, and the material that should be
used to produce the commemorative canister. Explain your reasoning. Then, present
your report to the class and the class can decide which report is the most convincing.
Be prepared to share your solutions and methods.
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Constructing a Net from a Model of a Geometric Solid
Geometric solids are bounded three-dimensional geometric
figures. The three dimensions are length, width, and
height. A net is a two-dimensional representation
of a geometric solid. A net can be cut out,
folded, and glued or taped to create a model of
a geometric solid. When constructing a net, it
may be helpful to make a sketch of each side or
face of the geometric solid first. Then, connect
each side in such a way that they share common
edges. When folded along these edges, the net
should be a model of the geometric solid.
Example
A net is sketched from the given model of a geometric solid.
Key Terms geometricsolids(14.1)
prototype(14.1)
edge(14.1)
face(14.1)
vertex(14.1)
net(14.1)
point(14.2)
linesegment(14.2)
polygon(14.2)
polyhedron(14.2)
regularpolyhedron(14.2)
congruent(14.2)
cube(14.2)
unitcube(14.2)
diameter(14.2)
surfacearea(14.2)
volume(14.2)
prism(14.3)
basesofaprism(14.3)
lateralfaces(14.3)
heightofaprism(14.3)
rectangularprism(14.3)
rightprism(14.3)
pyramid(14.5)
vertexofapyramid(14.5)
heightofapyramid(14.5)
slantheightofapyramid(14.5)
Chapter 14 Summary
Chapter 14 Summary • 989
Whoo! That was a tough
chapter and I know I made a lot of mistakes but you know, as Einstein said,
"A person who never made a mistake, never tried
anything new."
990. . . •. . . Chapter 14. . . Introduction.to.the.Third.Dimension
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Notice how the faces that share sides in the net share common edges in the model of the
cube. The other faces that share edges in the model are connected by the tabs.
top
left front right back
bottom
Calculating the Surface Area and Volume of Cubes
A polyhedron is a 3-dimensional solid that has polygons as faces. A regular polyhedron
has congruent regular polygons as faces and has congruent angles between all faces. A
cube is a regular polyhedron whose six faces are congruent squares. Surface area is the
total area of the 2-dimensional surfaces that make up a 3-dimensional object. The surface
area of a cube is calculated by determining the area of one face and then multiplying
that area by 6. Volume is the amount of space occupied by an object. To calculate the
volume of a cube, use the formula V 5 B 3 h, where V represents the volume of the cube,
B represents the area of the base of the cube, and h represents the height of the cube.
Example
7 cm
To determine the area of one face of the cube, multiply the length times the width.
Area of the base: 7 3 7 5 49 cm2
Surface area of the cube: 6 3 49 5 294 cm2
Use the formula V 5 B 3 h to calculate the volume of the cube.
V 5 B 3 h
5 49 3 7
5 343
The volume of the cube is 343 cm3.
When a cube’s dimensions are doubled, the volume of the resulting cube is eight times the
volume of the initial cube. If the given cube’s dimensions were doubled, the volume would
be 8 3343 52744 cm3.
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Calculating the Surface Area and Volume of Right Rectangular Prisms
A prism is a polyhedron with two parallel and congruent faces called bases. All other faces
are parallelograms and are referred to as lateral faces. A rectangular prism is a prism that
has rectangles as its bases. A right prism is a prism that has bases aligned one directly
above the other and has lateral faces that are rectangles. To calculate the surface area of a
right rectangular prism, calculate the area of each rectangular face and add each of these
areas together. As with cubes, the volume of a prism can be determined by using the
formula V 5 B 3 h, where V represents the volume of the prism, B represents the area of
the base of the prism, and h represents the height of the prism.
Example
7 cm
3 cm
2 cm
Surface area of the right rectangular prism:
2(7 32) 1 2(7 33) 1 2(2 33)
5 28 1 42 1 12
5 82 cm2
Area of the base of the prism: 7 32 5 14 cm2
Use the formula V 5 B 3 h to calculate the volume of the prism.
V 5 B 3 h
5 14 3 3
5 42
The volume of the prism is 42 cm3.
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Calculating the Surface Area and Volume of Prisms
A prism is a polyhedron with two parallel and congruent faces called bases. All other faces
are parallelograms and are referred to as lateral faces. The base of a prism can be any of a
variety of polygons. To calculate the surface area of a prism that does not have a rectangle
as a base, calculate the area of each face and add the areas together. The volume of a
prism can be determined by using the formula V 5 B 3 h, where V represents the volume
of the prism, B represents the area of the base of the prism, and h represents the height of
the prism.
Example
4.2 cm
5 cm
4.2 cm
5 cm
base
6 cm
5 cm
lateralface
To determine the area of the base, divide it into six congruent triangles and multiply the
area of one triangle by 6. Recall that the area formula for a triangle is A 5 1 __ 2
3b 3h,
where A represents the area of the triangle, b represents the length of the triangle’s base,
and h represents the height of the triangle.
Area of one triangular piece of the base: 1 __ 2
35 34.2 510.5 cm2
Area of the base of the prism: 6 310.5 563 cm2
Area of each lateral face: 635 530 cm2
Surface area of prism 5 Area of base 1 Area of lateral faces
5 63 1 (6 3 30)
5 63 1 180
5 243
The surface area of the prism is 243 cm2.
Use the formula V 5 B 3 h to calculate the volume of the prism.
V 5 B 3 h
5 243 3 6
5 1458 cm3
The volume of the prism is 1458 cm3.
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Calculating the Surface Area of Pyramids
A pyramid is a polyhedron with one base. The other faces of the pyramid are called lateral
faces and each lateral face is in the shape of a triangle. The number of lateral faces is
equal to the number of sides of the base. The vertex of a pyramid is the point at which all
lateral faces intersect. All of the pyramids associated with this chapter have a vertex that
is located directly above the center point of the base of the pyramid. The surface area is
determined by adding the areas of the base and each lateral face.
Example
8 cm
base
lateralface
13.2 cm
5.6 cm
To determine the area of the base, divide it into five congruent triangles and multiply the
area of one triangle by 5. Recall that the area formula for a triangle is A 5 1 __ 2
3b 3h,
where A represents the area of the triangle, b represents the length of the triangle’s base,
and h represents the height of the triangle.
Area of one triangular piece of the base: 1 __ 2
38 35.6 522.4 cm2
Area of the base of the pyramid: 5 322.4 5112 cm2
Area of each lateral face: 1 __ 2
38 313.2 552.8 cm2
Surface area of pyramid 5 Area of base 1 Area of lateral faces
5 112 1 (5 3 52.8)
5 112 1 264
5 376
The surface area of the pyramid is 376 cm2.
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Applying Surface Area Concepts to a Real World Situation
Many objects in the real world are geometric solids. People in a variety of jobs must apply
the surface area concepts studied in this chapter in order to solve real world problems
involving geometric solids.
Example
Horatio plans to construct steel grain bins in the shape of right rectangular prisms. The
steel sheeting he will use costs $3.50 per square foot. The top face of the grain bin will be
attached with three hinges to allow it to open and close. The cost of each hinge is $1.50.
How much will the material for one grain bin cost?
10 feet
3.5 feet
4 feet
Surface area of right rectangular prism: 2(10 3 3.5) 1 2(10 3 4) 1 2(4 3 3.5) 5 70 1 80 1
28 5 178 ft2
The cost of the steel sheeting for one grain bin is 3.50(178) 5 $623.00.
The cost of the hinges for one grain bin is 1.50(3) 5 $4.50.
The total material cost for one grain bin is $623.00 1 $4.50 5 $627.50.