Unit
6
Measurement
Unit Overview
In this unit your learning will be focused on
measurement and visualization of solid figures.
Essential Questions
?
How do two-dimensional
figures help you visualize
three-dimensional figures?
?
Why are geometric formulas
useful in solving real-world
problems?
Academic Vocabulary
© 2010 College Board. All rights reserved.
Add these words and others you encounter in this unit
to your vocabulary notebook.
lateral area
cylinder
cone
polyhedron
prism
face
edge
surface area
volume
EMBEDDED
ASSESSMENTS
This unit has 2 Embedded
Assessments. These embedded
assessments will allow you to
demonstrate your understanding of
isometric, top, bottom, and sides
views of three-dimensional figures,
similarity, and the formulas related
to solid figures.
Embedded Assessment 1
3-Dimensional Figures
p. 435
Embedded Assessment 2
Surface Area and Volume
p. 453
411
UNIT 6
Getting
Ready
a.
b.
6.
8 units
1. Solve the following equation for s.
1 r(q + s)
P = __
3
2. Identify the following two and three
dimensional figures.
c.
8 units
d.
e.
In the figure above, a circle is inscribed in a
square that has side length 8 units. Express the
shaded area of the figure in terms of π.
f.
The figure below is composed of three geometric
figures and should be used to respond to items 7
and 8.
Y
B
27
a
5
A
x
8
C
X
24
Z
3. Find the measure of sides a and x.
4. Name the pairs of angles that are congruent.
5. Give the area of a circle that has a
circumference of 15π units.
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
7. Name each of the figures and tell the formula
you would use to find its area.
8. Find the area of the composite figure.
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x
© 2010 College Board. All rights reserved.
y
Items 3 and 4 refer to the figure below.
ABC ∼ ZYX
Nets and Views of Solid Figures
What’s Your View?
SUGGESTED LEARNING STRATEGIES: Create Representations,
Use Manipulatives, Activate Prior Knowledge, Think/Pair/
Share; Self/Peer Revision. Interactive Word Wall
Len Oiler has gone into business for himself. He designs and builds tents
for all occasions. Len usually builds a model of each tent before he creates
the actual tent. As he builds the models, he thinks about two things: the
frame and the fabric that covers the frame. The patterns for Len’s first two
designs are shown by the nets below.
1. Build a frame for each model.
Figure 1
ACTIVITY
6.1
My Notes
MATH TERMS
A net is a 2-dimensional
representation for a
3-dimensional figure.
Figure 2
ACADEMIC VOCABULARY
2. A prism is a 3-dimensional solid with two congruent, polygonal
and parallel bases, whose other faces are called lateral faces. Knowing that prisms are named according to their base shape and whether
they are right or oblique, refer to the first tent frame you constructed
based on Figure 1.
prism
Faces are the polygons that
make up a 3-dimensional
figure. An edge is the
intersection of any two
faces. A vertex is a point
at the intersection of 3 or
more faces.
height
height
© 2010 College Board. All rights reserved.
a. What is the shape of the two congruent and parallel bases?
right prism
oblique prism
b. What is the shape of the three lateral faces?
MATH TERMS
c. What is the best name for the solid formed by the net in Figure 1?
d. Use geometric terms to compare and contrast right prisms and
oblique prisms.
In an oblique prism, the edges
of the faces connecting the
bases are not perpendicular
to the bases. In a right prism,
those edges are perpendicular
to the bases.
Unit 6 • Measurement
413
ACTIVITY 6.1
Nets and Views of Solid Figures
continued
What’s Your View?
My Notes
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Think/Pair/Share, Self/Peer Revision,
Interactive Word Wall
MATH TERMS
In an oblique pyramid, the
intersection of the lateral faces
(the top point of the pyramid)
is not directly over the center
of the base as it is in a right
pyramid.
height
right pyramid
height
oblique pyramid
3. A pyramid is a 3-dimensional solid with one polygonal base, whose
lateral faces are triangles with a common vertex. Knowing that
pyramids are named according to their base shape and whether they
are right or oblique, refer to the second tent frame you constructed
based on Figure 2 on the previous page.
a. What is the shape of the base?
b. What is the best name for the solid formed by the net in Figure 2?
Len publishes a brochure to help him “pitch” his tents. The brochures
include diagrams of the frames as isometric drawings, which are
three-dimensional drawings created on dot paper. The diagram below
is an example of an isometric drawing of a right rectangular prism.
414
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c. Use geometric terms to compare and contrast right pyramids and
oblique pyramids.
Nets and Views of Solid Figures
ACTIVITY 6.1
continued
What’s Your View?
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Think/Pair/Share, Group Presentation,
Self/Peer Revision
My Notes
4. Consider the figure drawn on dot paper on the previous page.
a. Explain why some of the segments that represent the edges are
solid and some are dashed.
Straight lines in a figure are
always drawn as straight lines
in an isometric dot diagram but
right angles in a figure are not
always depicted as 90° angles in
an isometric dot diagram.
b. Where are the right angles in the figure that are not drawn as
90° angles?
5. Where is the observer in relation to the solid discussed in Item 4?
6. Consider the figures in the My Notes column and then create isometric
drawings for the solids.
a. Assign whole number dimensions to the right rectangular prism
pictured in Figure 3.
Figure 3
© 2010 College Board. All rights reserved.
b. Assign whole number dimensions to the net of the right
rectangular prism in Figure 4.
7. The figure below right is a hexagonal pyramid. Locate the net for this
solid on the worksheets your teacher has given you.
T
a) List the vertices of the pyramid.
Figure 4
b) List the base edges.
F
c) List the lateral edges.
E
A
D
B
C
d) List the lateral faces.
Unit 6 • Measurement
415
ACTIVITY 6.1
Nets and Views of Solid Figures
continued
What’s Your View?
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Self/Peer Revision, Group Presentation
My Notes
8. Complete the table below. Nets for the pentagonal prism and pyramid
can be found on the worksheets your teacher has given you. All listed
figures are examples of polyhedra (singular: polyhedron).
ACADEMIC VOCABULARY
A polyhedron is a threedimensional figure
consisting of polygons
that are joined along their
edges.
Figure Name
Number of
Vertices
Total Number
of Edges
7
8
4
7
12
18
Triangular Prism
Rectangular Prism
Pentagonal Prism
Square Pyramid
Pentagonal Pyramid
MATH TERMS
In any polyhedron, the intersection of any two faces is called
an edge, and the vertices of the
faces are the vertices of the
polyhedron.
Number of
Faces
4
9. List any patterns that you observe in the table above.
10. Write a rule that expresses the relationship between F, V, and E.
Leonard Euler (1707–1783) was
one of the first mathematician
to write about the relationship
between the number of faces,
vertices and edges in polyhedrons.
Leonard Euler was not just a
“one-dimensional” man; he made
contributions to the fields of
physics and ship building, too.
11. Three views of the hexagonal pyramid are shown below.
a. Label each view as side, top, or bottom.
b. Explain the significance of the dashed segments in the views.
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CONNECT TO HISTORY
Nets and Views of Solid Figures
ACTIVITY 6.1
continued
What’s Your View?
SUGGESTED LEARNING STRATEGIES: Group Presentation,
Self/Peer Revision
My Notes
12. Sketch and label the bottom, top, and
side views of this rectangular prism.
13. If the bottom of the solid formed by
the net to the right is Face B, sketch
the top, front, and side views.
A
B
C
D
E
© 2010 College Board. All rights reserved.
14. Sketch the bottom, top, and side views of a square pyramid.
A cross section of a solid figure is the intersection of that figure and a plane.
Cross
section
Bottom of the solid
15. Below, are several cross sections of a hexagonal right pyramid.
Label each cross section as being parallel or perpendicular to the
base of the pyramid.
Unit 6 • Measurement
417
ACTIVITY 6.1
Nets and Views of Solid Figures
continued
What’s Your View?
My Notes
SUGGESTED LEARNING STRATEGIES: Group Presentation,
Self/Peer Revision, Activating Prior Knowledge,
Think/Pair/Share, Interactive Word Wall
16. Sketch several cross sections that are parallel to the base of a square
pyramid. Sketch several cross sections that are perpendicular to the
base of a square pyramid. What patterns do you observe?
cylinder
CONNECT TO AP
Right cylinders, cones, and
spheres are often called
solids of rotation because
they can be formed by rotating
a rectangle, triangle and semicircle around an axis. In calculus,
you will learn how to compute the
volume of solids of rotation using
a definite integral.
418
A cylinder is the set of all points in space that are a given distance (radius)
from a line, known as the axis. Bases are formed by the intersection of two
parallel planes with the cylinder. If the axis is perpendicular to each of the
bases, the cylinder is called a right cylinder.
17. Label the axis, radius, and height in the diagram of the right cylinder.
18. Describe the shape and size of the cross sections that are parallel to
the bases.
19. Sketch an oblique cylinder.
20. Len is working on a pattern for a right cylinder. Which base is the
correct size for this cylinder? Explain your answer.
SpringBoard® Mathematics with Meaning™ Geometry
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ACADEMIC VOCABULARY
Nets and Views of Solid Figures
ACTIVITY 6.1
continued
What’s Your View?
SUGGESTED LEARNING STRATEGIES: Group Presentation,
Think/Pair/Share, Activating Prior Knowledge
My Notes
21. Explain how the dimensions of the rectangle in the net on p. 418
relate to the cylinder once it is constructed.
A cone is the union of all segments in space that join
points in a circle to a point (called the vertex of the
cone) that is not coplanar with the circle. The base
of a cone is the intersection of a cone and a plane.
If the segment that joins the center of the circle
to the vertex is perpendicular to the plane that
contains the center, the cone is a right cone and this
segment is called the height of the cone. Otherwise,
the cone is oblique.
V
ACADEMIC VOCABULARY
cone
height
C
radius of
the base
© 2010 College Board. All rights reserved.
22. Compare and contrast the shape and size of the cross sections of a
right cone parallel to the base with the cross sections of a cylinder
that are parallel to its base.
23. Describe the cross section of a right cone that is perpendicular to the
base and contains the vertex of the cone. What is the best name for
this cross section? Explain how you know.
When the tent business gets slow, Len makes lampshades, which are formed
when a plane parallel to the base intersects the cone.
Unit 6 • Measurement
419
ACTIVITY 6.1
Nets and Views of Solid Figures
continued
What’s Your View?
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Group Presentation, Self/Peer Revision
My Notes
24. Len knows the height of the cone
is 30 inches and the radius is
9 inches. He needs to know the area
and circumference of the smaller
circle, which is the top of the shade.
(Assume the cross sections are
20 inches apart and parallel.)
V
Q
MATH TERMS
The section of a solid figure that
lies between two parallel planes
is called a frustum.
a. Label the center of the__
small
circle P, and draw in PQ.
Identify two similar triangles
in the figure. Explain why they
are similar.
20
C
A
b. Let r represent the radius of the small circle. Using corresponding
sides of the similar triangles, write a proportion and solve for r.
c. Find the circumference and area of the smaller cross-section.
25. Consider the cross sections of a sphere.
MATH TERMS
a. Describe the shape and size of the cross sections.
b. Where will the largest of these cross sections be located?
c. Where is the center of the largest possible cross section (called a great
circle) of a sphere located in relation to the center of the sphere?
26. The figure to the right is a sphere.
a. Write a definition for a chord of a sphere.
Draw and label a chord in the diagram.
b. Write a definition for the diameter of a
sphere. Draw and label a diameter in the diagram.
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A sphere is the set of all points
in space that are a given
distance (radius) from a given
point (center).
Nets and Views of Solid Figures
ACTIVITY 6.1
continued
What’s Your View?
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Group Presentation, Debriefing, Self/Peer
Revision
My Notes
c. Write a definition for a tangent (line or plane) to a sphere. Draw
and label a tangent line and a tangent plane in the diagram.
d. What must be true about a plane that is tangent to a sphere and
a radius of the sphere drawn to the point of tangency? Explain
your answer.
Len loves to cook outside on his new grill. The grill is spherical in shape,
and the radius of the sphere is 50 cm. The top and bottom are congruent
hemispheres, and there is a circular shelf inside and towards the bottom of
the grill for charcoal. This shelf is 30 cm. from the center of the grill.
27. Len wants to cover the shelf with aluminum foil so he needs to
calculate the area of the shelf. Use the diagram of the sphere below
to assist you with your responses.
C
A
© 2010 College Board. All rights reserved.
shelf
a. Draw the segment whose endpoints are the center of the sphere,
C, and the center of the shelf, B. Find the length of this segment.
b. Draw in the radius of the sphere that contains point A, which is
where the shelf is attached to the grill. Determine AC.
__
c. Complete ABC by drawing AB. What type of triangle is ABC?
__
d. How does AB relate to the circular shelf?
e. Calculate AB and the area of the shelf.
28. How far from the center of a sphere with radius 25 inches is the circle
whose circumference is 48π inches?
Unit 6 • Measurement
421
ACTIVITY 6.1
Nets and Views of Solid Figures
continued
What’s Your View?
My Notes
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Think/Pair/Share, Debriefing, Self/Peer
Revision, Interactive Word Wall, Group Presentation
29. The figure to the right is a right cone. A is the
vertex and C is the center of the base. If B is a
point on the base, then AB is called the slant
height of the cone. Draw ABC and calculate
the slant height if the diameter of the base is
6 cm and the height is 9 cm.
A
C
B
30. The figure below is a right rectangular prism with measures as shown.
B
a. Sketch ACD and find the
length of each side.
C
D
A
4 ft
E
G
8 ft
H
6 ft
c. Sketch ACH and find the length of each side.
d. Is ACH acute, obtuse, or right? Explain your answer.
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b. Sketch ACG and find the length of each side.
Nets and Views of Solid Figures
ACTIVITY 6.1
continued
What’s Your View?
SUGGESTED LEARNING STRATEGIES: Group Presentation,
Self/Peer Revision
My Notes
31. The figure below is a right square pyramid. Each side of the square
base has a length of 10 inches and each lateral edge has a length of
13 inches. Point C represents the center of the base.
__
__
__
a. Draw TM such that TM ⊥ PQ at
M. ___
Explain why M is the midpoint
of PQ.
T
13 in.
R
C
P
10 in.
b. TM is called the slant height of the
pyramid. Calculate TM.
Q
__
c. CT is the __
altitude of the pyramid. Sketch CMT and find the
length of CT .
__
© 2010 College Board. All rights reserved.
d. Find a right triangle in the pyramid that has CQ as one of its sides.
Calculate CQ.
32. The pyramid below has four congruent
equilateral faces. The length of each base
and lateral edge is 12 yards. Point T is the
center of WYZ.
__
a. Locate point M on WY such that XM is
the slant height. Find XM.
X
Z
MATH TERMS
Another name for a pyramid
with four congruent faces is a
regular tetrahedron.
12 yd
T
W
Y
__
b. Sketch TMY and find the length of the sides. (TM is the
apothem of WYZ.)
c. Sketch XTM and explain how to find the length of the altitude
of the pyramid.
Unit 6 • Measurement
423
ACTIVITY 6.1
Nets and Views of Solid Figures
continued
What’s Your View?
CHECK YOUR UNDERSTANDING
Write your
youranswers
answerson
onnotebook
notebookpaper.
paper.
Show
your work.
Show
your
5. The figure below is a cube with each edge
work.
length equal to 5 mm.
1. Consider the prism below.
A
C
B
__
__
a. AB is a diagonal of one face. Calculate AB.
b. AC is a diagonal of the cube. Calculate AC.
b. Does this drawing look like an isometric dot
figure? Explain.
2. Sketch and label the top, bottom and side
views of the figure shown above.
3. Sketch several cross sections that are
parallel to the bases of the triangular prism
below. Sketch several cross sections that
are perpendicular to the bases of the prism.
What patterns do you observe?
6. Find the area of the circle formed by the
intersection of a plane that is 24 mm from the
center of a sphere whose radius is 30 mm.
7. Mrs. Flowers is planning her annual garden
party. In case of rain, she has ordered a
large tent, which is a square right pyramid
as shown below.
V
E
S
C
B
4. MATHEMATICAL Use geometric terms to
R E F L E C T I O N compare and contrast
prisms and pyramids.
M
A
C represents the center of the
__base and M
represents the midpoint of AS. Mrs. Flowers
needs the height of the tent to be 25 ft. and
the base edge to be 120 ft. Calculate each of
the following.
a. the slant height
b. the length of a lateral edge
c. CM
d. AC
8. Find the slant height of a cone whose height
is 8 feet and whose diameter is 16 feet.
424
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a. Create a net for the figure.
Lateral Area and Surface Area
Exterior Experiences
ACTIVITY
6.2
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Create Representations
My
MyNotes
Notes
Brooke Taylor manages a group of tour guides and bush pilots; they call
their business Exterior Experiences. Brooke’s responsibilities include
booking the tours, purchasing supplies and equipment, advertising and
accounting. Brooke loves her job! The clients always come back delighted
with their vacation, and all of her guides are colorful people with expert
knowledge. Occasionally, she even gets to go along on a trip.
Brooke needs to order tents for a large group to be led into a
wilderness area to view grizzly bears. Their guide, Geronimo Cardano,
has been leading these tours for years. The tents need to vary in size but
have the same shape and color scheme. Geronimo wants each tent to
have four congruent faces in the shape of an equilateral triangle (regular
tetrahedron). He prefers this shape because the tent poles used to
construct the frame all have the same length.
ACADEMIC VOCABULARY
The surface area of a
three-dimensional figure
is the total area of all of its
surfaces.
1. Suppose the length of each edge of the tent is 8 ft.
a. Find the area of each face.
© 2010 College Board. All rights reserved.
b. Find the total surface area of
the four faces.
Recall the formula for finding the
area of an equilateral triangle,
given the__length of a side, s:
s2 √3
A = _____
4
2. Use the formula for calculating the area of an equilateral triangle to
derive a formula for the total surface area of one of these tents in
terms of the length of an edge, s.
3. Gino Leake Otare has created a fabric that is both water and puncture
resistant. He has received an order for 27.75 sq yd of this special
fabric for one of Geronimo’s tents. Find the dimensions of this tent.
Unit 6 • Measurement
425
ACTIVITY 6.2
Lateral Area and Surface Area
continued
Exterior Experiences
SUGGESTED LEARNING STRATEGIES: Visualization, Create
Representations
My Notes
Mike “The Pike” Rolle is a bush pilot, and he’s flying a group of Junior
Rangers to a remote lake for a week of camping and trout fishing. They
will be sleeping in tents that are in the shape of a triangular prism. The
net below is a pattern for one of the Junior Ranger tents. Locate the two
smaller right triangles at the bottom of the pattern. When they are zipped
together, they form a triangle that is congruent to the equilateral triangle
at the top of the pattern.
6
6
6
6
8
8
6
8
6
3
zipper
6
8
6
3
zippers
zippers
4. Find the area and perimeter of the equilateral triangular base.
6. Notice that the lateral faces of the prism form a larger rectangle.
a. What are the dimensions of this larger rectangle?
b. How do the dimensions of the larger rectangle relate to the
triangular base and the distance between the two bases?
ACADEMIC VOCABULARY
The lateral area of a solid
is the surface area of
the solid, excluding the
base(s).
426
c. The area of the largest rectangle is the sum of the areas of the
lateral faces, known as the lateral area. Find the lateral area of
this prism.
d. If h represents the distance between the two bases in a prism,
create a formula for finding the lateral area of a right prism in
terms of h and the perimeter of the base, p.
SpringBoard® Mathematics with Meaning™ Geometry
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5. Once the tent is constructed, what is the distance between the
two bases?
Lateral Area and Surface Area
ACTIVITY 6.2
continued
Exterior Experiences
SUGGESTED LEARNING STRATEGIES: Visualization, Create
Representations, Group Presentation
My Notes
7. The total surface area of a prism is the sum of the lateral area and the
area of the two bases.
a. Calculate the total surface area for one of the Junior Ranger tents.
b. Write a formula for calculating the total surface area, SA, in terms
of the lateral area, LA, and the base area, B.
8. Mike sleeps in a tent with a different shaped front and back than the
other tents, as shown below. Find the lateral area and the total surface
area of Mike The Pike’s tent (including the floor). Show your work.
10
5
5
5
10
7
© 2010 College Board. All rights reserved.
4
4
4
8
10
Avalanche Blanche Bombelli is leading a group on an over-night hike
to the Makame-Shiver Glacier. The winds are harsh (due to the high
elevation) so their tents cannot have any vertical sides. Blanche has
ordered tents in the shape of square right pyramids, one of which is
shown below.
Unit 6 • Measurement
427
ACTIVITY 6.2
Lateral Area and Surface Area
continued
Exterior Experiences
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Create Representations
My Notes
9. Let each side of the square base of the tent be 12, and the length of
each lateral edge be 10.
a. Determine the perimeter and the area of the base.
b. Draw one of the slant heights and calculate the area of one of the
triangular faces.
c. The lateral area of a pyramid is the sum of the areas of the lateral
faces. Calculate the lateral area of the pyramid.
d. If l represents the slant height of the pyramid, create a formula for
finding the lateral area, LA, of a right pyramid in terms of l and
the perimeter of the base, p.
10. The total surface area of a pyramid is the sum of the lateral area and
the area of the base.
a. Calculate the total surface area for the tent decribed in Item 9.
11. The regular tetrahedron at the beginning of this activity is a pyramid
with an equilateral triangular base. Use your formulas in items 9
and 10 to find the lateral area and total surface area for a regular
tetrahedron, each of whose edges is 8 feet in length.
TRY THESE A
a. Find the lateral and total surface
areas of a prism whose base is a right
triangle with legs 5 in. and 12 in. and
whose height is 3 in.
3
5
12
b. Find the lateral and total surface areas of a regular hexagonal pyramid
with base edge 6 cm and slant height 10 cm.
428
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b. Write a formula for calculating the total surface area, SA, of a right
pyramid in terms of the lateral area, LA, and the base area, B.
Lateral Area and Surface Area
ACTIVITY 6.2
continued
Exterior Experiences
SUGGESTED LEARNING STRATEGIES: Activating Prior
Knowledge, Create Representations
c. Find the height of a right prism
with lateral area 90 sq ft if the base
is a regular pentagon each of whose
sides is 4 ft.
My Notes
4
4
4
4
4
Gino Leake Otare makes cylindrical
stuff sacks from water and puncture
resistant fabric. The bases have a 6 inch
radius, and the height of each cylinder
is 15 inches.
12. Calculate the circumference and area of one of the bases of the stuff
sack. Leave your answers in terms of π.
© 2010 College Board. All rights reserved.
13. If it were laid out on a table, the lateral surface of the cylinder is in the
shape of which polygon? What are its dimensions?
14. How do the dimensions of the lateral surface relate to the circular
base and the height of the cylinder?
15. Determine the lateral area of each stuff sack.
16. Write a formula for calculating the lateral area, LA, of a right cylinder
in terms of the radius, r, the height, h, and π.
17. Calculate the total surface area of one of Gino’s stuff sacks.
18. Write a formula for calculating the total surface area, SA, of a cylinder
in terms of the lateral area, LA, and the base area, B.
Unit 6 • Measurement
429
ACTIVITY 6.2
Lateral Area and Surface Area
continued
Exterior Experiences
My Notes
Idaho Joe Lagrange guides tourists through National Parks on 4-wheeled
ATV’s. His tents are conical in shape to honor the indigenous people that
inhabited the region before the European settlers arrived. Each of Idaho
Joe’s teepees needs to be 10 feet in diameter and 12 feet high.
Gino Leake Otare charges by the square yard for the fabric that is
used to make tents. He would like to have a formula for calculating the
surface area for a teepee in terms of the dimensions.
Gino knows the formula for the area of the base of a cone, which
is a circle. It’s the lateral area that is troubling him. He begins his
investigation by examining several patterns for the curved surface of a
cone, the lateral surface.
19. Gino quickly realizes that each pattern is actually a sector of a circle.
Find the patterns for the curved surface of several cones on the
worksheet your teacher provided. Cut out each pattern and build the
lateral surface for each cone. Trace each circular base on a sheet of
paper and answer the questions below.
b. As the lateral area of the cone decreases, what happens to the
radius of the base?
c. As the lateral area of the cone decreases, what happens to the slant
height of the cone?
d. Which part of the cone corresponds to the radius of the sector?
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a. As the lateral area of the cone decreases, what happens to the
height of the cone?
Lateral Area and Surface Area
ACTIVITY 6.2
continued
Exterior Experiences
My Notes
Gino starts each pattern for the lateral surface with a large circle with
radius, l. Then, he cuts out a sector as shown below.
l
l
a
20. Use the figures above to respond to each of the following.
a. Write an expression for the area and circumference of the circle in
terms of l and π.
b. Write the ratio of the area of the sector, AS, to the area of the circle.
© 2010 College Board. All rights reserved.
c. Write the ratio of the length of the arc of the sector, a, to the
circumference of the circle.
d. Explain why the two ratios in Parts b and c can be used to create a
proportion.
e. Write the proportion from Part d, and solve this proportion for
AS. Simplify your answer.
f. How does a (the length of the arc of the sector) relate to the base
of the cone?
Unit 6 • Measurement
431
ACTIVITY 6.2
Lateral Area and Surface Area
continued
Exterior Experiences
My Notes
g. Write an expression for the lateral area of a cone, LA, in terms of l,
the radius of the base, r, and π.
h. Write a formula for calculating the total surface area, SA, in terms
of the lateral area, LA, and the base area, B.
21. Idaho Joe’s teepees are 10 feet in diameter and 12 feet high. Calculate
the lateral area and total surface area of each teepee.
TRY THESE B
a. Find the lateral area and the surface area for a cylinder whose height is
8 cm and radius is 9 cm.
b. Find the lateral area and surface area for a right cone
whose slant height forms a 45° angle with the base
whose radius is 8 mm.
CONNECT TO AP
In calculus, you will learn how
to derive the formula used to
calculate the surface area of a
sphere using limits or a definite
integral.
d. Idaho Joe is taking a group to ride ATV’s over sand
dunes in the desert. He needs large cool-down tents.
These tents are cylindrical in shape with a conical
top. Find the total surface area (including the floor)
of a tent if the diameter is 24 ft, the height of the
cylindrical portion is 6 ft and the height of the
conical portion is 5 ft.
Ryder Rivers leads groups of adventurers on river rafting trips. He wants
water proof containers for food and drinks that have spherical shapes.
These containers can be tied to the rafts and float along in the water. The
food and drinks inside the containers will stay cold because the river’s
primary sources are spring water and melting snow.
The spherical containers will be made of a shatter-proof plastic. The
formula for calculating the surface area of a sphere is: A = 4πr2 where r
represents the radius of the sphere.
432
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
c. Determine the height of a right cone whose lateral area is 136π ft2 and
base area is 64π ft2.
Lateral Area and Surface Area
ACTIVITY 6.2
continued
Exterior Experiences
SUGGESTED LEARNING STRATEGIES: Create Representations
My Notes
22. Calculate the surface area for one of the spherical containers if the
radius is 8 inches.
23. Determine the radius of a sphere whose surface area is 1256.64 cm2.
Round your answer to the nearest cm.
After a day of rafting, campers stay in a luxurious biosphere, which is
actually a hemisphere on a wooden platform.
24. Where would a plane have to intersect a sphere in order to form two
hemispheres? Explain your answer.
© 2010 College Board. All rights reserved.
25. Write a formula for calculating the surface area of a hemisphere in
terms of its radius, r. (Do not include the area of the circular cross
section.)
MATH TERMS
A hemisphere is half of a sphere.
26. Calculate the surface area of a hemisphere with radius 10 feet. Do not
include the area of the floor.
Ryder has placed bear-proof trash containers around the camp. These
containers have a cylindrical shape with a hemisphere-shaped lid as
shown below. Because he has to repaint them each spring, he needs to
know the surface area for each container.
27. Suppose the radius of the hemisphere and
cylinder is 1 foot, and the total height of the
container is 4 feet. Find the total surface area of
the trash container. (Include the bottom.)
Unit 6 • Measurement
433
ACTIVITY 6.2
Lateral Area and Surface Area
continued
Exterior Experiences
My Notes
28. A conical section has been carved into the flat surface of a hemisphere
as shown.
a. What information is needed to calculate the
total surface area?
b. Write the steps for calculating the total surface area of this figure.
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show
Write
paper.
Show
youryour work.
5.
work.
2. Determine the lateral and surface areas of the
right triangular right prism.
10 in.
9 in.
6 in.
8 in.
3. Calculate the lateral area and surface area of a
pyramid whose height is 8 ft and whose base is
a square with 12 ft sides.
4. Determine the slant height of a right pyramid
whose lateral area is 90 ft2 and whose base is a
regular hexagon with a 3-ft side.
434
SpringBoard® Mathematics with Meaning™ Geometry
formulas.
6. Determine the area of a label that covers the
lateral surface of a cylindrical can with height
40 mm and diameter 24 mm.
7. Calculate the lateral and surface areas of a
cone whose height is 12 in. and whose slant
height is 15 in.
8. Calculate the surface area of a sphere whose
diameter is 50 cm.
9. The figure below is a cone “topped” with a
hemisphere. Calculate the total surface area if
the radius of the cone and hemisphere is
10 cm and the height of the cone is 24 cm.
© 2010 College Board. All rights reserved.
1. Calculate the lateral and surface areas of a
prism whose height is 12 cm and whose base
is a rectangle with dimensions 5 cm by 7 cm.
MATHEMATICAL Compare and contrast the
R E F L E C T I O N pyramid and prism
3-Dimensional Figures
Embedded
Assessment 1
INFO NET
Use after Activity 6.2.
Below, you will find a net for a right solid. Do not cut out the net.
1. Write the name of the solid on one face of the net along with an isomorphic
drawing of the solid.
2. On another face, calculate the base area. Show your work and include the
formula(s) that you use.
3. On the third face, calculate the lateral area for the solid. Show your work and
include the formula that you use.
4. On the fourth face, calculate the total surface area for the solid. Show your
work and include the formula that you use.
5. On the fifth face, sketch and label two cross sections that are parallel to the base
and two cross sections that are perpendicular to the base.
6. On the remaining face, list the number of faces, vertices and edges for this solid
and determine whether or not this figure satisfies Euler’s Formula: F + V = E + 2.
15 cm
5 cm
5 cm
12 cm
© 2010 College Board. All rights reserved.
6 cm
5 cm
11 cm
Unit 6 • Measurement
435
3-Dimensional Figures
Use after Activity 6.2.
INFO NET
436
Exemplary
Proficient
Emerging
Math Knowledge
The student:
• Finds the correct
base area
and includes
the correct
formula(s) used.
• Finds the correct
lateral area for
the solid and
includes the
correct formula
used.
• Finds the correct
total surface
area for the solid
and includes the
correct formula
used.
The student:
• Uses the correct
formula to find
the base area
but makes a
computational
error.
• Uses the correct
formula to find
the lateral area
but makes a
computational
error.
• Uses the correct
formula to find
the surface area
for the solid
but makes a
computational
error.
The student:
• Uses an incorrect
formula to find
the base area.
• Uses an incorrect
formula to find
the lateral area.
• Uses an incorrect
formula to find
the surface area.
Problem Solving
The student:
• Identifies the
correct solid
represented by
the net.
• Correctly
determines
whether or
not the figure
satisfies Euler’s
formula.
The student:
• Indicates
knowledge of
Euler’s formula
but makes
an incorrect
determination.
The student:
• Does not identify
the correct solid.
• Does not
substitute
correctly into
Euler’s formula.
Representations
The student:
• Makes a correct
isomorphic
drawing.
• Sketches and
correctly labels
the four cross
sections.
The student:
• Makes an
isomorphic
drawing with a
minor error.
• Sketches and
correctly labels
only three of the
cross sections.
The student:
• Makes a
substantially
incorrect
isomorphic
drawing.
• Sketches and
correctly labels
one or two of the
cross sections.
Communication
The student lists
the correct number
of faces, vertices,
and edges for the
solid.
The student
lists two correct
amounts for the
faces, vertices, or
edges for the solid.
The student lists
only one correct
amount for the
faces, vertices, or
edges for the solid.
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
Embedded
Assessment 1
Volume
Cavalieri’s Kitchen
ACTIVITY
6.3
SUGGESTED LEARNING STRATEGIES: Visualization
My Notes
Cavalieri’s Kitchen Wows Customers with Volume of
Food and Fun
ith a menu that includes nothing but hash browns and pancakes,
Cavalieri’s Kitchen doesn’t appear to have much to offer most diners.
The steady flow of customers who have flooded the restaurant since
its opening two months ago definitely tell a different story. Owner, Greta
Carbo, sums up the premise behind the restaurant in a simple manner.
“There are two things that I do very well,” she says, decked out in her
signature white lab coat (complete with pocket protector). “I’m good with
mathematics, and I make a great pancake.”
When I asked how she was able to turn the two random qualities into a
successful business venture, she stated, “The key was to create a unique
experience that would make people want to come back, or better yet,
bring their friends and families back for more. You can get breakfast
anywhere,” she admits, “but you can’t get it anywhere else the way you
get it here.” As for the origin of the restaurant’s name, Carbo says it’s the
foundation for the establishment’s theme. “Basically, any solid can be
simulated by stacking a series of congruent or similar shapes. That’s
exactly what we do here.”
1. The regular Potato Prism consists of six patties of shredded potatoes
which have been molded into 5" × 5" squares. Before being served to
the customer, the patties are stacked atop one another to form a prism
as shown below.
© 2010 College Board. All rights reserved.
h
5
a. If the height of each patty is 0.5", determine the height of the
prism.
ACADEMIC VOCABULARY
b. We know that the volume of a prism is V = Bh, where B = the
area of the base. Determine the volume of the prism.
Volume is the number
of cubic units in a threedimensional figure.
Unit 6 • Measurement
437
ACTIVITY 6.3
Volume
continued
Cavalieri’s Kitchen
SUGGESTED LEARNING STRATEGIES: Visualization,
Quickwrite, Create Representations
My Notes
2. The price of all the items at Calavieri’s Kitchen is based strictly on the
volume of the food. Each cubic inch of food costs $0.05.
a. Based on this formula, calculate the menu price of the regular
Potato Prism.
b. Although the shape of the food is part of what makes the
restaurant unique, explain why the six patties would cost the
same regardless of how they were arranged on the plate.
3. The children’s menu includes similar items. However, the stacks are
arranged in pyramids.
1 Bh. Determine what the
b. The volume of a pyramid is V = __
3
volume of the Potato Pyramid should be based on the formula,
and explain why its actual volume may not be exactly the same.
c. Determine the menu price of the Potato Pyramid based the
volume in part b.
438
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
a. Draw a representation of a potato pyramid if the base is identical
to the prism and also includes six patties that are 0.5 thick.
Volume
ACTIVITY 6.3
continued
Cavalieri’s Kitchen
SUGGESTED LEARNING STRATEGIES: Visualization, Create
Representations
My Notes
The pancakes at Cavalieri’s Kitchen are arranged similarly to the potato
patties, however, they are served as cylinders and cones.
4. The Strawberry Cylinder consists of six strawberry pancakes, each
with a radius of 3 inches and a thickness of 0.5 inches.
a. Draw and label the Strawberry Cylinder in the My Notes column.
b. Determine the volume of the Strawberry Cylinder.
© 2010 College Board. All rights reserved.
c. Determine the menu price of the Strawberry Cylinder.
5. The Carrot Cone on the kid’s menu consists of six carrot pancakes.
The base pancake also has a radius of 3 inches, and each pancake
in the cone has a thickness of 0.5 inches. Based on what you know
about the volumes of cylinders and cones with the same base area and
height, what should be the menu price of the Carrot Cone? Draw and
label the Carrot Cone and explain your reasoning.
Unit 6 • Measurement
439
ACTIVITY 6.3
Volume
continued
Cavalieri’s Kitchen
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Questioning the Text, Summarize/Paraphrase/Retell
My Notes
In an exclusive to the SpringBoard Times Carbo informs me that she’s
experimenting with a new creation for the menu, the Blueberry Bubble.
“We’re very excited about it,” she beams. “It’s going to be a hemisphere of
blueberry pancakes that will absolutely melt in your mouth!” When asked
how the concocted confection will be made to follow the same mathematical
principles as the other menu items, Carbo assured us that it would. “It’s
actually much simpler than you would think.” she claims. “Just like the
cones on our menu, the hemisphere is really just a fractional portion of the
cylinders.”
“Here, I’ll show you,” Carbo says, “In each figure, the top of the shaded
disk is the same distance above the center of the figure.”
r
h
h
x
2r
x
h
x
2r
x
r
Each of the shaded disks in these figures is a cylinder, so the volume of
each disk is defined as V = πr2h. The volume of the disk in the cylinder
would be πr2h. The volume of the disk in the double cone is πx2h, and the
volume of the disk in the sphere is πr2h – πx2h.
By subtracting the volume of the disk in the double cone from the
volume of the disk in the cylinder, we get the volume of the disk in the
sphere. This relationship will hold true for the volumes of the entire
figures as well.
2r – 2 __
1 πr 2 · r
πr 2h · __
3
h
2 πr 3
= 2πr 3 – __
3
4 πr 3
= __
3
(
)
4 πr 3
Volume of a sphere = __
3
440
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
r
Volume
ACTIVITY 6.3
continued
Cavalieri’s Kitchen
SUGGESTED LEARNING STRATEGIES: Create Representations,
Visualization
My Notes
4 πr 3, what is the formula for the volume
6. If the volume of a sphere is __
3
of a hemisphere?
7. The Blueberry Bubble will have a base disk with a radius of 3 inches
and each of the six disks in the hemisphere will have a thickness of
0.5 inches.
© 2010 College Board. All rights reserved.
a. Draw a diagram for the Blueberry Bubble.
b. Based on the formula you derived in Question 6, what should be
the menu price of the Blueberry Bubble?
Unit 6 • Measurement
441
ACTIVITY 6.3
Volume
continued
Cavalieri’s Kitchen
SUGGESTED LEARNING STRATEGIES: Questioning the Text
My Notes
TRY THESE
a. The volume of a square pyramid is 243 in3. Determine the edge of the
base if the height of the pyramid is 27 in.
b. A right cylinder with diameter 2 inches has been carved out of the prism
shown below.
Diameter = 2 in.
12 in.
3 in.
8 in.
Determine the remaining volume of the prism. Justify your answer by
showing your work.
3 in.
2 in.
4 in.
If the ice cream melts completely, will it overflow the cone? Justify your
answer.
442
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
c. A spherical scoop of ice cream has been placed on top of a cone as
shown below.
Volume
ACTIVITY 6.3
continued
Cavalieri’s Kitchen
SUGGESTED LEARNING STRATEGIES: Prewriting, RAFT,
Visualization
My Notes
8. In an attempt to offer more healthy options to its customers,
Cavalieri’s Kitchen is expanding its menu to include the following
items. Create a page for the Cavalieri’s Kitchen menu that includes
graphic representations and prices for the items listed.
Potato Prism Lite: A stack of six square potato patties with side
lengths of 4 inches and a thickness of 0.25 inches.
Potato Pyramid Lite: A stack of six similar square potato patties. The
bottom patty has side lengths of 4 inches, and each patty has a thickness of 0.25 inches.
Strawberry Cylinder Lite: A stack of six round strawberry pancakes
with a radius of 2 inches and a thickness of 0.25 inches.
© 2010 College Board. All rights reserved.
Carrot Cone Lite: A graduated stack of six round carrot pancakes.
The bottom pancake has a radius of 2 inches, and each pancake has a
thickness of 0.25 inches.
Blueberry Bubble Lite: A graduated stack of six round blueberry
pancakes simulating a hemisphere. The bottom pancake has a radius
of 1.5 inches, and each pancake has a radius of 0.25 inches.
Unit 6 • Measurement
443
ACTIVITY 6.3
Volume
continued
Cavalieri’s Kitchen
CHECK YOUR UNDERSTANDING
Write your answers on notebook
notebook paper.
paper. Show
Show your work.
2. Calculate the volume of this right hexagonal
your work.
prism.
1. Calculate the volume of each figure.
a.
8 cm
9 cm
6 cm
8 cm
b.
8 ft
3. Determine the length of an edge of a cube
whose volume is 216 in.3.
4. If the bases of two cones shown below each
have radius 3 cm, will the volumes be the same?
Explain your reasoning.
5 ft
17 ft
c.
20 m
h
18 m
5. Four balls, each with radius 1 inch, fit snugly into
a 2" × 2" × 8" box. Is the total volume left over
inside the box greater than or less than the volume
of a fifth ball? Justify your answer.
d.
14 cm
6. MATHEMATICAL How would you find the
R E F L E C T I O N height of a cone if you know
the volume and the area of the base of the cone?
444
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
h
Similar Solids
Model Behavior
ACTIVITY
6.4
SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge,
Predict and Confirm
My Notes
Howie Shrinkett builds scale models for museums. He was commissioned
to build models of the solar system for the Hubble Planetarium, models
of the Egyptian Pyramids for the Ptolemy Natural History Museum and
models of the building that houses the Cubism Museum of Modern Art.
1. For the gift shop in the Cubism Museum of Modern Art, Howie created paper weights, key chains and planters. He began by investigating
the dimensions, surface area and volume of the cubes. Complete the
table below.
Item
Key
chain
Paper
weight
Planter
Edge length Area of Base Surface Area
(in square
(in inches)
(in square
inches)
inches)
Volume
(in cubic
inches)
1
3
9
2. Howie observed that as the length of an edge increases, so does the
resulting surface area.
© 2010 College Board. All rights reserved.
a. Make a conjecture about how the surface area changes when the
edge of a cube is tripled.
b. Use your conjecture and the information from the table to predict
the surface area of a cube whose edge is 27 inches. Then verify or
revise your conjecture from Part a.
3. Howie also observed that as the length of an edge increases, so does
the resulting volume.
a. Make a conjecture about how the volume changes when the edge
of a cube is tripled.
b. Use your conjecture and the information from the table to predict
the volume for a cube whose edge is 27 inches, and verify or revise
your conjecture from Part a.
Unit 6 • Measurement
445
ACTIVITY 6.4
Similar Solids
continued
Model Behavior
SUGGESTED LEARNING STRATEGIES: Predict and Confirm,
Quickwrite
My Notes
4. For the Hubble Planetarium, Howie created scale models of Mars.
Complete the table below for Howie’s spheres.
Radius
(in feet)
12
6
3
Surface Area
(in square feet)
Volume
(in cubic feet)
5. Howie observed that as the radius decreases, so does the resulting
surface area.
a. Make a conjecture about how the surface area changes when the
radius is cut in half.
b. Use your conjecture to predict the surface area of a sphere whose
radius is 1.5 inches, and verify or revise your conjecture from Part a.
a. Make a conjecture about the way the volume changes when the
radius is cut in half.
b. Use your conjecture to predict the volume of a sphere whose radius
is 1.5 inches, and verify or revise your conjecture from Part a.
7. Suppose the values in the table for the sphere had been given in yards.
Explain how your conjectures in Items 5a and 6a would change?
446
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
6. Howie also observed that as the radius decreases, so does the
resulting volume.
Similar Solids
ACTIVITY 6.4
continued
Model Behavior
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Visualization, Activating Prior Knowledge
My Notes
Howie recalled that similar two-dimensional and three-dimensional
figures are those that have the same shape but not necessarily the same
size. Does this imply that all cubes are similar to all other cubes, and that
all spheres are similar to all other spheres? Explain below.
As Howie begins his pyramid project for the Ptolemy Museum, he
realizes that he needs more than one dimension to determine a scale
model. The scale models must be geometrically similar.
8. Which of the pyramids below are similar? Explain.
10 in.
© 2010 College Board. All rights reserved.
12 in.
8 in.
20 in.
17.5 in.
Pyramid 1
Pyramid 2
14 in.
Pyramid 3
9. Howie has built a square pyramid. The base edge is 10 centimeters
and the height is 12 centimeters. He knows that he needs the height of
the second pyramid to be 24 centimeters. Determine the scale factor
of the small pyramid to the larger pyramid.
10. Determine the length of the base edge of the larger pyramid.
Unit 6 • Measurement
447
ACTIVITY 6.4
Similar Solids
continued
Model Behavior
SUGGESTED LEARNING STRATEGIES: Visualization
11. Complete the table below for Howie’s pyramids.
Height
of
pyramid
(in in.)
Length
of the
base edge
(in in.)
Slant
Height
(in in.)
Area
Perimeter
Surface
Lateral
of the
of the
Area
Volume
Area
Base
Base
(in in.) (in sq. in.) (in sq. in.) (in sq. in.) (in cu. in.)
Pyramid 1
Pyramid 2
Ratio of
Pyramid
1
_________
Pyramid 2
12. Howie also designs models for cylindrical drums. Scale drawings for the drums are given below. Complete the
table below, and justify whether or not the two cylinders are similar.
4 cm
3 cm
2 cm
Cylinder 1
Radius
(in cm)
Circumference of
base (in cm)
Cylinder 1
Cylinder 2
Ratio of
Cylinder
1
_________
Cylinder 2
448
SpringBoard® Mathematics with Meaning™ Geometry
Cylinder 2
Surface Area
(in square cm)
Volume
(in cubic cm)
© 2010 College Board. All rights reserved.
6 cm
Similar Solids
ACTIVITY 6.4
continued
Model Behavior
SUGGESTED LEARNING STRATEGIES: Quickwrite,
Self/Peer Revision
My Notes
13. Complete the table below for any pair of similar solids with the scale
factor a:b.
Ratio of
Ratio
sides/edges
of
or other
corresponding perimeters
measurements
Similarity
Ratios
Ratio
of
areas
Ratio
of
volumes
a
__
b
a. Write a conjecture about the ratio of corresponding perimeters in
any pair of similar solids.
© 2010 College Board. All rights reserved.
b. Write a conjecture about the ratio of corresponding areas in any
pair of similar solids.
c. Write a conjecture about the ratio of the volumes of any similar
solid.
Unit 6 • Measurement
449
ACTIVITY 6.4
Similar Solids
continued
Model Behavior
SUGGESTED LEARNING STRATEGIES: Visualization,
Self/Peer Revision
My Notes
© 2010 College Board. All rights reserved.
14. Draw and label two solids whose corresponding dimensions do not
have the same scale factors. Based on the conjectures you made in
Item 13, verify that the two figures are not similar to one another.
450
SpringBoard® Mathematics with Meaning™ Geometry
Similar Solids
ACTIVITY 6.4
continued
Model Behavior
SUGGESTED LEARNING STRATEGIES: Visualization
My Notes
15. A tetrahedron has four faces, each of which is an equilateral triangle.
The diagram below shows two tetrahedrons. Suppose the tetrahedron
on the left has 6-inch edges and the tetrahedron on the right has
8-inch edges.
a. What is the ratio of the surface area of the smaller tetrahedron to
the surface area of the larger tetrahedron?
b. What is the ratio of the volume of the smaller tetrahedron to the
volume of the larger tetrahedron?
© 2010 College Board. All rights reserved.
16. Two similar cones have volumes of 125π in.3 and 64π in.3.
a. Determine the scale factor for these cones.
b. If the total area of the larger cone is 75π in.2, calculate the total
area of the smaller cone.
Unit 6 • Measurement
451
ACTIVITY 6.4
Similar Solids
continued
Model Behavior
CHECK YOUR UNDERSTANDING
© 2010 College Board. All rights reserved.
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show
Write
paper.
Show
youryour work.
2. You purchased a scale model of the Statue of
work.
Liberty. The scale factor is 1:456. The model
is 8 inches high and has a surface area of
1. A pyramid has a height of 6 cm and base edge
0.052 ft2.
8 cm. A similar pyramid has a height of 9 cm.
a. What is the height of the Statue of Liberty?
a. Find the scale factor of the smaller pyramid
b. What is the surface area of the Statue of
to the larger pyramid.
Liberty?
b. What is the ratio of the base edges?
3. MATHEMATICAL If you know that the ratio
c. Find the base edge of the larger pyramid.
R E F L E C T I O N of the edges of two cubes
d. What is the ratio of the lateral areas?
5 , how do you find the ratio for the area of
is __
3
e. What is the ratio of the volumes?
the faces of the cubes and the ratio for the
volumes of the cubes?
f. If the volume of the larger pyramid is
3
130.5 cm , what is the volume of the smaller
pyramid?
452
SpringBoard® Mathematics with Meaning™ Geometry
Surface Area and Volume
Embedded
Assessment 2
ACTION-PACKED MEASUREMENTS
Use after Activity 6.4.
Play-Tel Toys has entered talks to create action figures for the latest Hollywood
blockbuster film. The film’s main character, a robot named Arigato, is the focus of
the marketing campaign.
In order to do a cost analysis for creating the action figures, Play-Tel designers must
determine the surface area and volume of the robot’s body parts. Although the final
shapes will be crafted in much greater detail, each body part will first be molded
into the solids as indicated in the diagram to the right.
1. Complete the table below based on the dimensions
provided in the diagram.
3 cm
Neck?
8 cm
Surface Area
8 cm
Volume
Spherical Head
1 cm
1 cm
10 cm
Right Cylinder Torso
5 cm
Right Cylinder Arm
Square Prism Leg
2. The designers need to know the total surface area of the
entire robot in order to determine painting costs. If it costs
$0.02 per square centimeter, determine the cost to paint
each action figure.
12 cm
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3. The designers also need to determine the total amount
of plastic to be used in the molds for each robot. If each cubic centimeter of
plastic costs $0.03, determine the total cost to mold each action figure.
12 cm
4 cm
4 cm
4. The designers are discussing the best figure to use for the neck of the robot.
They have narrowed their decision to three options: a right circular cone with
a radius of 1 cm and a height of 2 cm, a square pyramid with base edge 2 cm
and a height of 2 cm, and a right hexagonal pyramid with base edges of 1 cm
and a height of 2 cm. Which figure will cost the least amount to produce?
Justify your answer.
Unit 6 • Measurement
453
Embedded
Assessment 2
Surface Area and Volume
Use after Activity 6.4.
ACTION-PACKED MEASUREMENTS
454
Exemplary
Proficient
Emerging
Math Knowledge
#1, 2, 3, 4, 5
The student:
• Completes
the table with
eight correct
measurements. (1)
• Calculates the
correct total
surface area of
the robot and the
cost to paint each
figure. (2)
• Calculates the
correct total
amount of plastic
and the cost of
the plastic needed
to make each
robot. (3)
• Uses the correct
volume formulas
for the three
options. (4)
• Uses the scale
factor to find
the correct total
production
cost. (5)
The student:
• Completes
the table with
only six or
seven correct
measurements.
• Calculates the
correct total
surface area
of the robot
but makes a
computational
error when
calculating the
cost.
• Calculates the
correct total
amount of plastic
but makes a
computational
error when
calculating the
cost.
• Uses the
correct volume
formulas for the
three options
but makes a
computational
error.
• Uses the scale
factor but makes
a computational
error.
The student:
• Completes
the table
with at least
three correct
measurements.
• Calculates an
incorrect surface
area.
• Calculates an
incorrect volume.
• Uses the
incorrect volume
formula for some
of the options.
• Does not use
the scale factor
correctly.
Communication
#4
The student writes
a mathematically
accurate justification
for the figure
chosen. (4)
The student writes
an incomplete
justification
that contains no
mathematical
errors.
The student writes
a justification
that contains
mathematical
errors.
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5. Now that full-scale production has begun on the action figures, the company
wants to create life-size versions of the robots to display in stores across the
country. If the scale factor of the action figure to a life-size model is 1:9,
determine the total production cost of creating one life-size model.
Practice
ACTIVITY 6.1
Brian’s hiking boots came in an unusual box. The
top and bottom were right triangles, the legs of
which measured 12 and 16 inches. The height of the
box was 10 inches.
UNIT 6
5. The figure below is a square pyramid with height
10 cm and base edge 20 cm. Calculate each of
the following:
P
S
12
Q
R
16
1. If the box is sitting on the largest lateral face,
sketch the top, bottom, and side views.
a.
b.
c.
d.
the slant height
the length of a lateral edge
the length of the apothem of the base
the length of the radius of the base
ACTIVITY 6.2
2. Calculate each of the following for the box in
Item 1.
PQ
PR
SQ
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3. Use geometric terms to describe the cross
sections (parallel and perpendicular to the base)
of a cylinder with height 8 inches and radius
10 inches.
6. Find the lateral area and surface area of a
right prism whose height is 10 ft and whose
base
__
is a rhombus with diagonals 12 ft and 12 √3 ft.
7. Find the lateral area of a right pyramid whose
slant height is 18 mm and whose base is a square
with area 121 mm2.
4. Find the area of a circle formed by the
intersection of a plane 10 inches away from the
center of a sphere whose radius is 20 inches.
Unit 6 • Measurement
455
Practice
UNIT 6
8. The figure below is regular hexagonal right
prism with a regular hexagonal pyramid carved
into a base of the prism. Find the total surface
area of the figure if the height of the prism and
pyramid is 8 cm, and the length of one side
of the hexagonal base of the prism and the
pyramid is 6 cm.
12. A cylinder has a volume of 200 in.3. Determine
the volume of a cone whose radius and height are
equal to that of the cylinder.
13. Determine the volume of the right triangular
prism.
6
8
15
14. A US gold dollar has a radius of 26.5 mm and a
thickness of 2 mm. Calculate the total volume of
gold in a stack of 25 gold dollars.
9. Determine the lateral area of a pipe that is 12 ft
long and 4 inches in diameter.
10. Calculate the surface area of the cylindrical
rocket and “nose cone” if the slant height of the
nose cone is 8 ft.
15. A cylindrical tank with a diameter of 8 feet and a
height of 14 feet is being filled with water. If the
water is entering the tank at a rate of 2 cubic feet
per second, how long will it take for the tank to
be half full?
16. Determine the volume of the figure below.
6
6 cm
ACTIVITY 6.3
11. A square pyramid has a volume of 196 cm3.
Determine the height of the pyramid if the edges
of the base measure 7 cm.
12 cm
3 cm
3 cm
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20
Practice
17. The cylinder shown below has been capped with
a hemisphere at each end. The height of the
cylinder is 10 cm. The radius of each hemisphere
is 2 cm. Calculate the volume of the figure.
UNIT 6
19. A pentagonal prism has a total surface area
of 256 cm2. Explain what happens to the total
surface area if the dimensions of the prism
are doubled.
20. Cone A and Cone B have a similarity ratio of 2:3.
What is the volume of Cone A if the volume of
Cone B is 36 cubic units?
ACTIVITY 6.4
18. A cylinder has a radius of 4 cm and a height of
9 cm. A similar cylinder has a radius of 6 cm.
22. You purchased a scale model of the Eiffel
Tower. The scale factor is 1:1180. The model
is 10 inches high and has a surface area of
1.7 ft2.
a. What is the height of the Eiffel Tower?
b. What is the surface area of the Eiffel Tower?
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a. Find the scale factor of the smaller cylinder
to the larger cylinder
b. What is the ratio of the circumferences of
the bases?
c. What is the ratio of the lateral areas of
the cylinders?
d. What is the ratio of the volumes of
the cylinders?
e. If the volume of the smaller cylinder is
144π cm3, what is the volume of the larger
cylinder?
21. A sphere with diameter 5 ft is being painted.
How much more paint would be needed to paint
a sphere that has 4 times the diameter?
Unit 6 • Measurement
457
UNIT 6
Reflection
An important aspect of growing as a learner is to take the time to reflect on
your learning. It is important to think about where you started, what you have
accomplished, what helped you learn, and how you will apply your new knowledge in
the future. Use notebook paper to record your thinking on the following topics and to
identify evidence of your learning.
Essential Questions
1. Review the mathematical concepts and your work in this unit before you write
thoughtful responses to the questions below. Support your responses with
specific examples from concepts and activities in the unit.
How do two-dimensional figures help you visualize three-dimensional figures?
Why are geometric formulas useful in solving real-world problems?
Academic Vocabulary
2. Look at the following academic vocabulary words:
lateral area
prism
cone
face
polyhedron
surface area
cylinder
edge
volume
Choose three words and explain your understanding of each word and why each is
important in your study of math.
Self-Evaluation
3. Look through the activities and Embedded Assessments in this unit. Use a table
similar to the one below to list three major concepts in this unit and to rate your
understanding of each.
Unit
Concepts
Is Your Understanding
Strong (S) or Weak (W)?
Concept 2
Concept 3
a. What will you do to address each weakness?
b. What strategies or class activities were particularly helpful in learning the
concepts you identified as strengths? Give examples to explain.
4. How do the concepts you learned in this unit relate to other math concepts and to
the use of mathematics in the real world?
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Concept 1
Math Standards
Review
Unit 6
1. An artist created a solid copper sculpture in the shape of a
1. Ⓐ Ⓑ Ⓒ Ⓓ
sphere with a radius of 3 cm. He was not pleased with his work,
so he melted it down and started to pour all of the copper into
a cone-shaped mold with a radius of 3 cm and a height of 6 cm.
Which of the following is true?
1 of the cone.
A. The copper will fill about __
2
3
__
B. The copper will fill about of the cone.
4
C. The copper will approximately fill the cone with almost none
left over.
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D. There is more copper than can fit into the cone, so there will
be extra copper.
2. When designing their house, Susie wants her closet to be much
bigger than Sam’s. On the blueprints, Sam’s closet is 1 cm by
3 cm. Susie made her closet 3 cm by 9 cm. What is the ratio of
the area of Sam’s closet to Susie’s closet assuming they are built
according to the blueprints?
2.
3. The XYZ cereal company is designing a package for a new
product. A right rectangular prism-shaped package stacks best
on store shelves. The cereal will all fit into a prism that measures 4 cm long, 3 cm wide, and 5 cm high. The art department
is designing graphics for the package and needs to know how
much room they have to work with. What is the total surface
area, in square centimeters, of the package for the cereal?
3.
‒
○
⊘⊘⊘⊘
• ○
• ○
• ○
• ○
• ○
•
○
⓪⓪ ⓪ ⓪ ⓪ ⓪
①① ① ① ① ①
②② ② ② ② ②
③③ ③ ③ ③ ③
④④ ④ ④ ④ ④
⑤⑤ ⑤ ⑤ ⑤ ⑤
⑥⑥ ⑥ ⑥ ⑥ ⑥
⑦⑦ ⑦ ⑦ ⑦ ⑦
⑧⑧ ⑧ ⑧ ⑧ ⑧
⑨⑨ ⑨ ⑨ ⑨ ⑨
‒
○
⊘⊘⊘⊘
• ○
• ○
• ○
• ○
• ○
•
○
⓪⓪ ⓪ ⓪ ⓪ ⓪
①① ① ① ① ①
②② ② ② ② ②
③③ ③ ③ ③ ③
④④ ④ ④ ④ ④
⑤⑤ ⑤ ⑤ ⑤ ⑤
⑥⑥ ⑥ ⑥ ⑥ ⑥
⑦⑦ ⑦ ⑦ ⑦ ⑦
⑧⑧ ⑧ ⑧ ⑧ ⑧
⑨⑨ ⑨ ⑨ ⑨ ⑨
Unit 6 • Measurement
459
Math Standards
Review
Unit 6 (continued)
4. The spa in Ester’s new house is in the shape of a cylinder with a
diameter of 8 feet. The depth of the spa is 5 feet. Ester wants to
Solve
know how much water the spa will hold when filled to a depth
Explain
of 3 feet.
Read
Part A: Draw a sketch and label the dimensions of the spa and
the height of the water.
Part B: Determine the volume of water, in cubic feet, needed to
fill the spa to the 3-foot depth. (Use π = 3.14)
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Part C: Water weighs about 62 pounds per cubic foot.
Determine the weight of the water to the nearest pound. Show
your work or write a written explanation to justify your answer.
460
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