3-D Geometry
Intro to 3-D Solids
Class Work
1. For each of the following name the edges, faces, and vertices.
a.
b.
c.
2. For each solid, name the lateral edges and base edges.
a.
b.
c.
3. Consider the figures in question 2. How many vertices, edges, and faces does each one
have? Prove that Euler’s Theorem (V + F = E + 2) holds true for each solid.
4. Draw the cross-section indicated
a.
b.
c
5. Describe the cross-section of a hexagonal prism given that the plane of intersection is
a. Between and parallel to the bases
b. Contains corresponding diagonals of the bases
c. Intersects all of the faces
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PARCC-type question:
6. Ann buys a block of clay for a Geometry project. The block is shaped like a cylinder with a
base area of 48𝜋 𝑐𝑚2 , and the height is three times the radius. Ann decides to cut the block
of clay into two pieces. She places a wire across the diameter of the circular base as shown
in the figure. Then, she pulls the wire straight down to create 2 congruent chunks of clay.
Ann wants to keep one chunk of clay for later use. To keep that chunk from drying out, she
wants to place a piece of plastic sheeting on the surface she exposed when she cut through
the cylinder. Describe the newly exposed two-dimensional cross section, and find its area.
Round your answer to the nearest whole square inch. Show your work.
Intro to 3-D Solids
Homework
7. For each of the following the name edges, faces, and vertices.
a.
b.
c.
8. For each solid, name the lateral faces and base(s).
a.
b.
c.
9. Consider the figures in question 8. How many vertices, edges, and faces does each one
have? Prove that Euler’s Theorem (V + F = E + 2) holds true for each solid.
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10. Draw the cross-section indicated
a.
b.
c.
11. Describe the cross-sections of a cone given the that the plane of intersection is
a. Parallel to the base
b. Oblique to the base but not intersecting the base
c. Intersects the base at 2 points
PARCC-type question:
12. Rick buys a block of clay for a Geometry project. The block is shaped like a rectangular
prism with length edges of 12 in, width edges of 6 in, and height edges of 5 in. Rick decides
to cut the block of clay into two pieces. He places a wire across the diagonal of the front
face of the prism as shown in the figure. Then, he pulls the wire straight back to create 2
congruent chunks of clay.
Rick wants to keep one chunk of clay for later use. To keep that chunk from drying out, he
wants to place a piece of plastic sheeting on the surface he exposed when he cut through
the prism. Describe the newly exposed two-dimensional cross section, and find its area.
Round your answer to the nearest whole square inch. Show your work.
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Views & Drawings of 3-D Solids
Class Work
13. Sketch the front, side (right side), and top views of the figure.
a.
b.
14. Draw the solid given the top, front, and side (right side) views
a.
b.
Homework
15. Sketch the front, side (right side), and top views of the solid.
a.
b.
16. Draw the solid given the top, front, and side (right side) views
a.
b.
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Surface Area of a Prism
Class Work
17.
A jewelry store buys small boxes in which to wrap items that they sell. The diagram
below shows one of the boxes. Find the lateral area and the surface area of the box to the
nearest whole number.
18. Find the lateral area and surface area of the hexagonal prism.
19. Find the surface area of the composite space figure.
20. Consider the prism shown below.
a. Draw a net for the prism and label all dimensions.
b. Use the net to find the surface area of the prism.
21. Find the lateral area and surface area of the triangular prism.
a.
b.
c.
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Surface Area of a Prism
Homework
22. Draw the net and label all of the dimensions for the prism. Find the lateral area and surface
area of the rectangular prism.
23. Find the lateral area and surface area of the regular pentagonal prism.
24. Find the surface area of the composite space figure.
A 5x6x8 box with a triangular prism removed.
(Triangle is equilateral)
25. Find the lateral area and surface area of the triangular prisms.
a
b.
Surface Area of a Cylinder
Class Work
26. The radius of the base of a cylinder is 39 in. and its height is 33 in. Find the surface area of
the cylinder in terms of .
27. Find the Lateral Area and Surface Area of the Cylinder
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28. Find the surface area of the composite figure.
PARCC-type Questions:
29. Andrew is planning to cover the lateral surface of a large cylindrical garbage can with
decorative fabric for a theme party. The can has a diameter of 3 feet and a height of 3.5
feet. How much fabric does he need if he covers the lid but not the bottom of the can?
30. Jalissa wants to paint just the sides of a cylindrical pottery vase that has a height of 35 cm
and a diameter of 12 cm. Find the number of square centimeters she will need to paint.
Explain the method you would use to find the lateral area.
31. A washer is a cylindrical solid with a smaller cylinder removed from the center and then
dipped in a special coating. If the diameter of the washer is 4”, the diameter of the hole is
1“, and height ¼ “, find the surface area of the washer.
Surface Area of a Cylinder
Homework
32. Find the Lateral Area and Surface Area of the Cylinder
a
b.
33. Find the surface area for the composite figure.
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PARCC-type Questions:
34. Andrew is planning to cover the lateral surface of a large cylindrical swimming pool with
decorative fabric for a theme party. The pool has a radius of 9 feet and a height of 4 feet.
How much fabric does he need?
35. Jasmin wants to paint just the sides of a cylindrical pottery vase that has a height of 48 cm
and a diameter of 19 cm. Find the number of square centimeters she will need to paint.
36. A washer is a cylindrical solid with a smaller cylinder removed from the center and then
dipped in a special coating. If the diameter of the washer is 3”, the diameter of the hole is
½“, and height is ¼ “, find the surface area of the washer.
Surface Area of a Pyramid
Class Work
37. Find the surface area of the pyramid.
38. Find the slant height, lateral area and surface area of the pyramid.
39. Find the slant height, lateral area and surface area of each square pyramid.
a.
b.
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40. Find the surface area of the pyramid shown below.
41. Find the surface area of the composite figure.
42. A rectangular pyramid fits exactly on top of a rectangular prism. The prism has a length of
15 cm, a width of 5 cm, and a height of 7 cm. The pyramid has a height of 13 cm. Find the
surface area of the composite figure.
Surface Area of a Pyramid
Homework
43. A regular hexagonal pyramid has base edges of 48 cm and a slant height of 26 cm. Find
the lateral area and surface area.
44. Find the slant height, lateral area and surface area of each square pyramid.
a.
b.
45. Find the surface area of the pyramid shown below. The base of the pyramid is a regular
octagon.
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46. Find the Surface Area of the composite figure.
Surface Area of a Cone
Class Work
47. Find the slant height of the cone.
48. Find the surface area of the cone in terms of  .
a.
b.
49. Find the surface area of a conical grain storage tank that has a height of 30 meters and a
diameter of 14 meters.
50. The lateral area of a right cone is 40𝜋 ft2, find the height of the cone if slant height is 10ft.
51. The surface area of a right cone is 55𝜋 cm2, find the radius if the slant height is 6cm.
Surface Area of a Cone
Homework
52. Find the slant height of the cone with height 7 and radius 6.
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53. Find the Lateral Area and surface Area of the cone.
a.
b.
54. The lateral area of a cone is 558 cm2. The radius is 31 cm. Find the slant height.
55. The lateral area of a right cone is 30𝜋 ft2, find the height of the cone if slant height is 8ft.
56. The surface area of a right cone is 30𝜋 cm2, find the radius if the slant height is 7cm.
Volume of Prisms
Class Work
57. Concrete can be purchased by the cubic yard. How much will it cost to pour a slab 18 feet
by 18 feet by 4 inches for a patio if the concrete costs $41.00 per cubic yard?
58. Find the volume of the rectangular prism
a.
b.
59. Find the volume of the triangular prism
a.
b.
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60. Find the volume.
a.
b.
Volume of Prisms
Homework
61. A jewelry store buys small boxes in which to wrap items that they sell. The diagram below
shows one of the boxes. Find the volume of the box.
62. Find the volume of the Rectangular Prism.
63. Find the volume of the box w/ triangular prism removed.
64. Find the volume of the Triangular Prism
a.
b.
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65. Find the volume.
Volume of Cylinders
Class Work
66. The radius of the base of a cylinder is 39 in. and its height is 33 in. Find the volume of the
cylinder.
67. A cylinder has a volume of 271.4 cubic inches and a base diameter of 12 in. Find the height
of the cylinder.
68. Find the volume of the cylinder.
69. Find the volume of the composite figure.
70. Denise wants to use a cylindrical garbage can for recycling. The can has a diameter of 4
feet and a height of 2.5 feet. How many cubic feet of recycling will fit into the garbage can?
71. A washer is a cylindrical solid with a smaller cylinder removed from the center. The
diameter of the washer is 3”, the diameter of the hole is ½”, and height ¼ “. Find the volume
of the washer.
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Volume of Cylinders
Homework
72. Find the volume of the cylinder.
a.
b.
73. Find the volume of the composite figure.
74. Denise is going to a potluck dinner party and needs to use a cylindrical container for her
potato salad. The container has a diameter of 15 inches and a height of 7 inches. How many
cubic inches of potato salad can Denise fit into the container to take to the pot luck dinner?
75. A washer is a cylindrical solid with a smaller cylinder removed from the center. If the
diameter of the washer is 4” and the diameter of the hole is 1“, and height ¼ “, find the
volume of the washer.
Volume of Pyramids
Class Work
76. Find the volume of the pyramid.
a.
b.
77. Find the volume of the composite figure.
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78. A rectangular pyramid fits exactly on top of a rectangular prism. The prism has a length of
15 cm, a width of 5 cm, and a height of 7 cm. The pyramid has a height of 13 cm. Find the
volume of the composite figure.
PARCC-type Question:
79. The table shows the approximate measurements of the Pyramid of the Sun in Mexico and
the Bent Pyramid in Egypt.
Pyramid
Length (meters)
Width (meters)
Height (meters)
Pyramid of the Sun
225
225
75
Bent Pyramid
188.6
188.6
101.1
Approximately, what is the difference between the volume of the Pyramid of the Sun and the
volume of the Bent Pyramid?
a. 68,103 cubic meters
b. 105,582 cubic meters
c. 200,752 cubic meters
d. 7,392,998 cubic meters
Volume of Pyramids
Homework
80. A square pyramid has base edges of 48 cm and a slant height of 26 cm. Find its volume.
81. Find the volume of the pyramid.
a.
b.
82. Find the volume of the composite figure.
83. A rectangular pyramid fits exactly on top of a rectangular prism. The prism has a length of
12 cm, a width of 6 cm, and a height of 8 cm. The pyramid has a height of 10 cm. Find the
volume of the composite figure.
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PARCC-type Question:
84. The table shows the approximate measurements of the two notable pyramid shaped
buildings in the United States: the Luxor Hotel in Las Vegas, Nevada and the Pyramid Arena
in Memphis, Tennessee.
Pyramid
Length (feet)
Width (feet)
Height (feet)
Luxor Hotel
600
600
350
Pyramid Arena
591
591
321
Approximately, what is the difference between the volume of the Luxor Hotel and the volume
of the Pyramid Arena?
a. 1,123,804 cubic feet
b. 4,626,933 cubic feet
c. 13,880,799 cubic feet
d. 79,373,067 cubic feet
Volume of Cones
Class Work
85. Find the volume of the cone in terms of  .
a.
b.
c.
86. A conical grain storage tank has a height of 30 meters and a diameter of 14 meters. Find the
capacity of the storage tank. Round the answer to the nearest square meter.
87. If a cone has height 6 ft and volume 54 ft3, find the radius of the base.
88. The vertex of a right cone has a 20o angle and the slant height is 10 cm. Find the volume of
the cone.
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PARCC-type Question
89. Geometryville Farms is building a silo to store the grain that has a cylindrical base and a
cone-shaped top. The laws of Geometryville say that the silo must have a maximum width
of 18 feet and a maximum height of 25 feet.
Truck cars are used to transport the grain in loads that are 9 feet tall, 8 feet
wide and 27.25 feet long. Geometryville wants to be able to store 3
truckloads of grain. Determine the height of the cylinder, ℎ1 , and the height
of the cone, ℎ2 , that Geometryville Farms should use in the design. Show
that your design will be able to store at least 3 truckloads of grain.
Volume of Cones
Homework
90. The lateral area of a cone is 558 cm2. The radius is 5 cm. Find the volume of the cone to
the nearest tenth.
91. Find the volume of the cone.
92. If a cone has height 9 ft and volume 75 ft3, find the radius of the base.
93. The vertex of a right cone has a 40o angle and the slant height is 12 cm. Find the volume of
the cone.
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PARCC-type Question
94. Geometryville Farms is building a silo to store the food that has a cylindrical base and a
cone-shaped top. The silo must have a maximum width of 16 feet and a maximum height of
22 feet.
Truck cars are used to transport the food in loads that are 9 feet tall, 8 feet
wide and 27.25 feet long. Geometryville wants to be able to store 2
truckloads of food. Determine the height of the cylinder, ℎ1 , and the height of
the cone, ℎ2 , that Geometryville Farms should use in the design. Show that
your design will be able to store at least 2 truckloads of food.
Surface Area & Volume of Spheres
Class Work
95. The equator of Earth is approximately 25,000 miles. What is the diameter of the Earth?
96. The cross section of a sphere taken 4 units from the center of the sphere has radius 6.
What is the radius of the sphere?
97. The cross section of a sphere taken 7 units from the center of the sphere has an area of
9𝜋u2. What is the radius of the sphere?
98. A new dome-shaped storage shed is a hemisphere with height 10 yds. What is the area of
the floor space?
99. A basketball has a circumference of 28”, what is the diameter of the ball?
100. Find the surface area of the sphere with the given dimension.
a. Radius = 60 m
b. Diameter = 24 cm
c. Circumference = 13 mm
101.
Find the surface area of a sphere that has a great circle with circumference of 13 mm.
Round to the nearest tenth.
102.
A balloon has a surface area of 200 cm2. Find the radius of the balloon.
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103.
Find the surface area of the sphere.
a.
b.
104. Three balls are packaged in a cylindrical container as shown below. The balls just touch
the top, bottom, and sides of the cylinder. The diameter of each ball is 7 cm.
a. What is the radius of the cylinder?
b. What is the height of the cylinder?
c. What is the total surface area of the three balls?
105.
Find the volume of the sphere with the given dimension. Leave your answer in terms of
.
a. Radius = 60 m
b. Diameter = 14 cm
c. Circumference = 13 mm
106.
A balloon has a surface area of 200 cm2. Find the volume of the balloon.
107. Find the volume of the sphere.
a.
b.
108. Three balls are packaged in a cylindrical container as shown below. The balls just touch
the top, bottom, and sides of the cylinder. The diameter of each ball is 7 cm.
a. What is the volume of the cylinder? Explain your method for finding the
volume.
b. What is the total volume of the three balls? Explain your method for finding
the total volume.
c. What percent of the volume of the container is occupied by the three balls?
Explain how you would find the percent.
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Surface Area & Volume of Spheres
Homework
109. The diameter of Jupiter is approximately 89,000 miles. How long is Jupiter’s equator?
110.
The cross section of a sphere taken 8 units from the center of the sphere has radius 7.
What is the radius of the sphere?
111.
The cross section of a sphere taken 5 units from the center of the sphere has an area of
6u2. What is the radius of the sphere?
112.
A new dome-shaped storage shed is a hemisphere with height 8 yds. What is the area
of the floor space?
113.
A baseball has a circumference of 9”, what is the diameter of the ball?
114.
Find the surface area of the sphere with the given dimension. Leave your answer in
terms of .
a. Radius = 45 m
b. Diameter = 16 cm
c. Circumference = 27 mm
115.
A ball has a surface area of 35.5 cm2. Find its radius.
116.
Find the surface area of the sphere.
a.
b.
117. Three balls are packaged in a cylindrical container as shown below. The balls just touch
the top, bottom, and sides of the cylinder. The diameter of each ball is 9 cm.
a. What is the radius of the cylinder?
b. What is the height of the cylinder?
c. What is the total surface area of the three balls?
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118. Find the volume of the sphere with the given dimension. Leave your answer in terms of
.
a. Radius = 45 m
b. Diameter = 16 cm
c. Circumference = 27 mm
133.
119.
A ball has a volume of 15.5 cm3. Find the ball’s surface area.
120.
a.
Find the volume of the sphere.
b.
121. Three balls are packaged in a cylindrical container as shown below. The balls just touch
the top, bottom, and sides of the cylinder. The diameter of each ball is 13 cm.
a. What is the volume of the cylinder? Explain your method for finding the
volume.
b. What is the total volume of the three balls? Explain your method for finding
the total volume.
c. What percent of the volume of the container is occupied by the three balls? Explain
how you would find the percent.
Cavalieri’s Principle
Class Work
122. The following solids have the same volumes. Find the value of x.
a.
b.
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123.
Determine whether Cavalieri’s Principle can be used to compare the volumes of any of
the solids. Explain your reasoning.
PARCC-type Questions:
124. Ann buys a block of clay for a Geometry project. The block is shaped like a cylinder with
a base area of 49𝜋 𝑐𝑚2 , and the height is three times the radius. Ann decides to cut the
block of clay into two pieces. She places a wire across the diameter of the circular base
as shown in the figure. Then, she pulls the wire straight down to create 2 congruent
chunks of clay.
Ann wants to reshape one chunk of clay to make a set of clay cubes. She wants each
cube to have a side length measurement of 5 cm. Find the maximum number of cubes
that Ann can make from the chunk of clay. Show your work.
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125.
Two rectangular prisms, each with a square base and a height of 20 cm, are shown.
Which statements about prisms E and F are true? Select all that apply.
a. If x < y, the area of the horizontal cross section of prism E is greater than the
horizontal cross section of prism F.
b. If x < y, the area of the horizontal cross section of prism E is equal to the
horizontal cross section of prism F.
c. If x < y, the area of the horizontal cross section of prism E is less than the
horizontal cross section of prism F.
d. If x > y, the volume of prism E is greater than the volume of prism F, because the
base area of prism E is greater than the base area of prism F.
e. If x > y, the volume of prism E is equal to the volume of prism F, because the
base area of prism E is equal to the base area of prism F.
f. If x > y, the volume of prism E is less than the volume of prism F, because the
base area of prism E is less than the base area of prism F.
Cavalieri’s Principle
Homework
126. The following solids have the same volumes. Find the value of x.
a.
b.
127.
Determine whether Cavalieri’s Principle can be used to compare the volumes of any of
the solids. Explain your reasoning.
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PARCC-type Questions
128. Two cylinders, each with a radius of 4 cm, are shown.
Which statements about Cylinders G and H are true? Select all that apply.
a. If x > y, the area of the vertical cross section of cylinder G is greater than the
vertical cross section of cylinder H.
b. If x > y, the area of the vertical cross section of cylinder G is equal to the vertical
cross section of cylinder H.
c. If x > y, the area of the vertical cross section of cylinder G is less than the vertical
cross section of cylinder H.
d. If x = y, the volume of cylinder G is greater than the volume of cylinder H,
because the height of cylinder G is greater than the height of cylinder H.
e. If x = y, the volume of cylinder G is equal to the volume of prism F, because the
height of cylinder G is equal to the height of cylinder H.
f. If x = y, the volume of cylinder G is less than the volume of prism F, because the
height of cylinder G is less than the height of cylinder H.
129.
Rick buys a block of clay for a Geometry project. The block is shaped like a rectangular
prism with length edges of 12 in, width edges of 6 in, and height edges of 5 in. Rick
decides to cut the block of clay into two pieces. He places a wire across the diagonal of
the front face of the prism as shown in the figure. Then, he pulls the wire straight back
to create 2 congruent chunks of clay.
Rick wants to reshape one chunk of clay to make a set of cylinders. He wants each
cylinder to have a radius of 1.5 in. and a height of 3 in. Find the maximum number of
cylinders that Rick can make from the chunk of clay. Show your work.
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Similar Solids
Class Work
130. Determine whether each pair of solids is similar or not. If similar, find the ratio of
similitude.
a.
b.
131.
132.
133.
The ratio of the slant heights of 2 similar pyramids is 2 to 5.
a. What is the ratio of their heights?
b. What is the ratio of their surface areas?
c. What is the ratio of their volumes?
The ratio of surface areas of 2 similar solids is 16 to 9.
a. What is the ratio of their heights?
b. What is the ratio of their lateral areas?
c. What is the ratio of their volumes?
The ratio of surface areas of 2 similar solids is 8 to 1.
a. What is the ratio of their heights?
b. What is the ratio of the area of their bases?
c. What is the ratio of their weights (made of the same materials)?
Homework
134. Determine whether each pair of solids is similar or not. If similar, find the ratio of
similitude.
a.
b.
135.
136.
137.
The ratio of the slant heights of 2 similar cones is 6 to 7.
a. What is the ratio of their radii?
b. What is the ratio of their surface areas?
c. What is the ratio of their volumes?
The ratio of surface areas of 2 similar solids is 25 to 4.
a. What is the ratio of their heights?
b. What is the ratio of their lateral areas?
c. What is the ratio of their volumes?
The ratio of surface areas of 2 similar solids is 27 to 125.
a. What is the ratio of their heights?
b. What is the ratio of the area of their bases?
c. What is the ratio of their weights (made of the same materials)?
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Review: 3-D Geometry
1. A right square pyramid has base edges of 6 and a height of 5. Find the slant height.
a. 3.3
b. 4
c. 5.8
d. 7.8
2. How many base edges does an oblique hexagonal prism have?
a. 6
b. 7
c. 12
d. 14
3. Find the number of vertices in the polyhedron if it contains 20 triangular faces and 12
pentagonal faces.
a. 30
b. 32
c. 60
d. 120
4. Find the lateral area of the right regular pyramid represented by the net.
5. The surface area of a box with length 5 cm and width 4 cm is 76 cm2. Find the height.
6. Find the surface area of a right cone with radius 3 and slant height 4.
7. A cross-section of a sphere, which is 6 units form the center of the sphere, has a
circumference of 8𝜋 units. Find the surface area of the sphere.
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8. Find the volume of the right triangular prism.
9. Find the volume of the cylinder with height 6 ft and radius 4 ft.
10. Find the volume of the right square pyramid with base edges of 8 and slant height of 10.
11. The surface areas of 2 similar spheres have the ratio of 9 to 4. If the volume of the larger
sphere is 405 u3, what is the volume of the smaller sphere?
a. 120 u3
b. 160 u3
c. 180 u3
d. 270 u3
12. Two rectangular prisms, each with a square base and a height of 30 cm, are shown.
30 cm
30 cm
Which statements about prisms E and F are true?
a. If x = y, the area of the horizontal cross section of prism E is greater than the
horizontal cross section of prism F.
b. If x = y, the area of the horizontal cross section of prism E is equal to the
horizontal cross section of prism F.
c. If x = y, the area of the horizontal cross section of prism E is less than the
horizontal cross section of prism F.
d. If x < y, the volume of prism E is greater than the volume of prism F, because the
base area of prism E is greater than the base area of prism F.
e. If x < y, the volume of prism E is equal to the volume of prism F, because the
base area of prism E is equal to the base area of prism F.
f. If x < y, the volume of prism E is less than the volume of prism F, because the
base area of prism E is less than the base area of prism F.
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13. Find the surface area and volume of the composite figure below if the cone has a slant
height of 10 inches, radius of 6 inches, and the cylinder has the same height as the cone.
14. In Geometryville’s Pentagon Park, they have a play tunnel shown in the figure below. It has
a diameter length of 42 inches and a height of 14 feet. This year, Geometryville will hire
some workers to paint the outside of the play tunnel with 3 new coats of paint. How much
paint do the workers need?
Extended Constructed Response
1. Consider the net of a figure made of cubes with edge lengths of 4 cm.
a. Sketch a 3-dimensional drawing of the figure.
b. What is the surface area of the solid?
c. What is the volume of the solid?
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2. The Geometryville Aquarium is building a new tank space for its Pacific fish shown in the
figure below. The laws say that the dimensions of the tank must have a maximum length of
16 feet, a maximum width of 12 feet and a maximum height of 18 feet.
Salt water comes in cylindrical containers that measure 10 feet high and have a diameter of
8 feet. Determine the heights of the aquarium that should be used in the design. Show that
your design will be able to store at least 4 cylindrical containers of water.
Geometry – 3-D Geometry
~29~
NJCTL.org
Answer Key
1.
a.)Edges:
b.) Edges:
̅̅̅̅
̅̅̅̅
𝑉𝑍, ̅̅̅̅
𝑉𝑌,
𝐴𝐵, ̅̅̅̅
𝐵𝐶 ,
̅̅̅̅, 𝑉𝑊
̅̅̅̅̅ ,
̅̅̅̅ , 𝐷𝐸
̅̅̅̅ ,
𝑉𝑋
𝐶𝐷
̅̅̅̅̅
̅̅̅̅
̅̅̅̅
𝑊𝑍, ̅̅̅̅̅
𝑋𝑊 ,
𝐷𝐹 , 𝐴𝐹 ,
̅̅̅̅
̅̅̅̅
̅̅̅̅
𝑋𝑌, ̅̅̅̅
𝑌𝑍;
𝑄𝑅 , 𝑅𝑆,
̅̅̅̅, 𝑇𝑈
̅̅̅̅,
Faces:
𝑆𝑇
̅̅̅̅
̅̅̅̅
VZY,
𝑈𝑉 , 𝑉𝑄 ,
̅̅̅̅
̅̅̅̅
VXY,
𝐴𝑉 , 𝐵𝑄 ,
VWX,
̅̅̅̅ , 𝐷𝑆
̅̅̅̅,
𝐶𝑅
VWZ,
̅̅̅̅, ̅̅̅̅
𝐸𝑇
𝑈𝐹 ;
WXYZ;
Faces:
Vertices:
ABCDEF,
V, W, X,
QRSTUV,
Y, Z
CRSD,
DSTE,
ETUF,
FUVC,
ABQR;
Vertices: A,
B, C, D, E,
F, Q, R, S,
T, U, V
c.) Edges:
̅̅̅,
̅̅̅̅ , ̅𝑇𝑆
𝑇𝑅
̅̅̅̅, ̅̅̅
𝑆𝑇
𝐽𝐾 ,
̅
̅̅̅̅
𝐽𝐿, 𝐾𝐿,
̅̅̅̅
𝑅𝐿, ̅̅̅̅
𝑆𝐾 ,
̅̅̅;
𝑇𝐽
Faces:
RST, JKL,
TRLJ,
TSKJ,
RSKL;
Vertices:
J, K, L, R,
S, T
a.) Base
b.) Base
edges:
edges:
̅̅̅̅, 𝐴𝑇
̅̅̅̅,
̅̅̅̅, 𝐵𝐶
̅̅̅̅ ,
𝑃𝐴
𝐴𝐵
̅̅̅̅, 𝑁𝐸
̅̅̅̅ ,
̅̅̅̅ , 𝐷𝐸
̅̅̅̅ ,
𝑇𝑁
𝐶𝐷
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
𝐸𝑃, 𝐺𝐵,
𝐸𝐹 , 𝐹𝐺 ,
̅̅̅̅ , 𝐶𝐷
̅̅̅̅ ,
̅̅̅̅
𝐵𝐶
𝐺𝐴
̅̅̅̅
̅̅̅̅ , 𝐹𝐺
Lateral
𝐷𝐹
edges:
Lateral
̅̅̅̅
edges:
𝑉𝐴, ̅̅̅̅
𝑉𝐵,
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅ ,
𝑃𝐺 , 𝐴𝐵,
𝑉𝐶 , 𝑉𝐷
̅̅̅̅ , 𝑁𝐷
̅̅̅̅ ,
̅̅̅̅ , 𝑉𝐹
̅̅̅̅ ,
𝑇𝐶
𝑉𝐸
̅̅̅̅
̅̅̅̅
𝐸𝐹
𝑉𝐺
3. 2a: V = 10; E = 15; F = 7:
V+F=E+2
10 + 7 = 15 + 2
17 = 17
2b: V = 8; E = 14; F = 8
V+F=E+2
8 + 8 = 14 + 2
16 = 16
2c: V = 8; E = 12; F = 6
V+F=E+2
8 + 6 = 12 + 2
14 = 14
c.) Base
edges:
̅̅̅̅,
̅̅̅̅̅, 𝑋𝑄
𝑊𝑋
̅̅̅̅̅, ̅̅̅̅̅̅
𝑄𝑀
𝑀𝑊 ,
̅̅̅̅
𝑁𝑃, ̅̅̅̅
𝑃𝑌,
̅̅̅̅
𝑌𝑍, ̅̅̅̅
𝑍𝑁
Lateral
edges:
̅̅̅̅̅
𝑀𝑁, ̅̅̅̅̅
𝑊𝑍,
̅̅̅̅
𝑋𝑌, ̅̅̅̅
𝑄𝑃
2.
Geometry – 3-D Geometry
~30~
4. a) Square
b) Ellipse
c) Circle
5. a) Hexagon
b) Rectangle
c) Octagon
6. Sample Answer: The shape of the
exposed surface is a rectangle. The
width of the rectangle is equal to the
diameter of the circle. Since the
circle’s area is 48𝜋 𝑐𝑚2 , we can
calculate the length of the radius
48𝜋 = 𝜋𝑟 2
48 = 𝑟 2
𝑟 = √48 = 4√3 𝑐𝑚 ≈ 6.93 𝑐𝑚
This means that the height of the
cylinder (length of the rectangle),
which is 3 times the radius, is equal
to 12√3 𝑐𝑚 ≈ 20.78 𝑐𝑚 and the
diameter (width of the rectangle) is
equal to 8√3 𝑐𝑚 ≈ 13.86 𝑐𝑚. To
calculate the area of the rectangle,
multiply the diameter and the height
of the cylinder.
12√3(8√3) = 96(3) = 288 𝑐𝑚2
7.
a) Edges: b)
Edges: c) Edges:
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅̅
𝐴𝐵, 𝑇𝐶 ,
𝐴𝐵, 𝐵𝐶 ,
𝑀𝑄 , ̅̅̅̅̅̅
𝑀𝑊 ,
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅̅
𝑁𝐷 , 𝐸𝐹 ,
𝐶𝐷 , 𝐷𝐸 ,
𝑊𝑋, ̅̅̅̅
𝑋𝑄 ,
̅̅̅̅ , 𝐴𝑇
̅̅̅̅,
̅̅̅̅
̅̅̅̅ ,
̅̅̅̅, 𝑃𝑌
̅̅̅̅,
𝑃𝐺
𝑁𝑃
𝐸𝐺 , 𝐹𝐺
̅̅̅̅, 𝑁𝐸
̅̅̅̅ ,
̅̅̅̅, 𝐴𝑉
̅̅̅̅ ,
̅̅̅̅
𝑇𝑁
𝑌𝑍, ̅̅̅̅
𝑍𝑁,
𝐺𝐴
̅̅̅̅
̅̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅̅
𝑀𝑁, ̅̅̅̅̅
𝑊𝑍,
𝐸𝑃, 𝑃𝐴,
𝐵𝑉 , 𝐶𝑉 ,
̅̅̅̅
̅̅̅̅
̅̅̅̅, ̅̅̅̅
̅̅̅̅ , 𝐸𝑉
̅̅̅̅ ,
𝐵𝐶 , 𝐶𝐷 ,
𝑋𝑌
𝑄𝑃
𝐷𝑉
̅̅̅̅ ,
̅̅̅̅
̅̅̅̅ , 𝐹𝐺
̅̅̅̅ , 𝐺𝑉
Faces:
𝐷𝐹
𝐹𝑉
̅̅̅̅
QMWX,
Faces:
𝐺𝐵
ABCDEFG, NPYZ,
Faces:
QXYP,
ABV, BCV,
ATNEP,
MWZN
CDV, DEV,
BCDFG,
Vertices:
EFV, GFV,
ABCT,
Q, P, M.
AGV
TCDN,
Vertices: A, N, W, X,
NDFE,
Y, Z
B, C, D, E,
EFGP,
F, G, V
PGBA
Vertices:
A, B, C,
D, E, F,
G, N, P
NJCTL.org
8.
a.)Lateral
faces:
AVQB,
BQRC,
CRSD,
DSTE,
ETUF,
FUVC
Bases:
ABCDEF,
QRSTUV
b.)Lateral
faces:
VWZ,
CZY,
VYX,
VXW
Bases:
WXYZ
c.)Lateral
faces:
TJLF,
TJKS,
RLKS
Bases:
TRS, JKL
13. a
b
14. a
8a: V = 12; E = 18; F = 8
V+F=E+2
12 + 8 = 18 + 2
20 = 20
8b: V = 5; E = 8; F = 5
V+F=E+2
5+5=8+2
10 = 10
8c: V = 6; E = 9; F = 5
V+F=E+2
6+5=9+2
11 = 11
10. a) Rectangle
b) Pentagon
c) Trapezoid
11. a) Circle
b) Ellipse
c) Parabola
12. Sample Answer: The shape of the
exposed surface is a rectangle. The
width of the rectangle is equal to the
width of the prism, which is 6 in. The
length of the exposed rectangle is the
diagonal (d) of the front and/or back
face of the prism. To find this length,
I’m going to use Pythagorean
Theorem.
52 + 122 = 𝑑2
25 + 144 = 𝑑2
169 = 𝑑2
𝑑 = √169 = 13 𝑖𝑛.
To calculate the area of the
rectangle, multiply the width of the
prism by the length of the diagonal
that we just calculated.
13(6) = 78 𝑖𝑛2
b
9.
Geometry – 3-D Geometry
~31~
15. a
b
16. a
b
17. LA = 69.72 cm2
SA = 265.72 cm2
18. LA = 192 u2
SA = 275.14 u2
19. 232 cm2
20. a) Nets will vary
b) 184 cm2
NJCTL.org
21. a) LA = (51 + 17√5)m2 = 89.01 m2
SA = (53 + 17√5) m2 = 91.01 m2
b) LA = 945 u2
SA = 1,139.86 u2
c) LA = 322.14 m2
SA = 332.14 m2
22. Nets will vary; LA= 416 ft2; SA=488 ft2
23. LA = 330 u2
SA = 453.87 u2
24. 262.54 u2
25. a) LA = 63 m2 & SA = 105.44 m2
b) LA = 650 ft2 & SA = 707.24 ft2
26. SA = 5,616in2
27. LA = 252cm2
SA = 350 cm2
28. 416.75 m2
29. 33 ft2
30. LA = 1,319.47 cm2; used dh
31. 27.49 in2
32. a) LA = 179.7 cm2
SA = 3,204.4 cm2
b) LA = 1,130.97 m2
SA = 1,639.91 m2
33. 375.48 mm2
34. 1,017.9 ft2
35. 2,865 cm2
36. 16.49 in2
37. SA = 95 m2
38. ℓ = 2√10 𝑚𝑚
LA = 50.6 mm2
SA = 66.6 mm2
39. a) ℓ = 5 𝑢𝑛𝑖𝑡𝑠, LA = 80 u2
SA = 144 u2
b) ℓ = 9.49 𝑢𝑛𝑖𝑡𝑠, LA = 278.9 u2
SA = 495 u2
40. SA = 1,008 + 294√3 𝑖𝑛2
≈ 1,517.22 𝑖𝑛2
41. SA = 40.66 ft2 (∆𝑠 are not congruent)
42. 628.6 in2
43. LA = 3,744 cm2; SA = 9729.97 cm2
44. a) ℓ = √61 𝑢𝑛𝑖𝑡𝑠, LA = 187.45 u2
SA = 331.45 u2
b) ℓ = 12.02 𝑢𝑛𝑖𝑡𝑠, LA = 479.39 u2
SA = 876.99 u2
45. SA = 936.32 cm2
46. 1,654.45 in2
47. ℓ = 21.47 cm
48. a) 60 ft2
b) 314.7 cm2
Geometry – 3-D Geometry
~32~
49. 831.39 m2
50. 9.2 ft
51. 5 cm
52. 9.21 units
53. a) LA = 252 ft2 & SA = 396ft2
b) LA = 90.48 u2 & SA = 140.74 u2
54. 5.73 cm
55. 7.07 ft
56. 3 cm
57. $164
58. a) 576 ft3
b) 308.88 cm3
59. a) 130 m3
b) 2045.99 u3
c) 17 m3
60. a) 180 cm3
b) 681.31 u3
61. 228 cm3
62. 25.5 in3
63. 231.34 cm3
64. a) 715.45 ft3
b) 63.65 m3
65. 332.55 u3
66. 157,685.96 in3
67. h=2.4 in
68. 882 cm3
69. 625.8 m3
70. 31.4 ft3
71. 1.72 in3
72. a) 1620m3
b) 450u3
73. 850.36 mm3
74. 1,237 cm3
75. 2.95 in3
76. a) 32 mm3
b) 110.85 u3
77. 18.75 ft3
78. 850 cm3
79. A
80. 1,214.3 cm3
81. a) 54.49 m3
b) 73.41 u3
82. 4,356 in3
83. 816 cm3
84. B
85. a) 157.7 ft3
b) 37,699.11 m3
c) 1989.68 cm3
86. 1539.38 m3
87. 2.9 ft
NJCTL.org
88. 3.11 cm3
89. Sample Answer: Assuming that the
truck cars are rectangular prisms,
each truck holds 1,962 cubic feet of
grains (27.25 x 9 x 8 = 1,962). Three
truck cars hold 5,886 cubic feet of
grains. The volume of the silo equals
the volume of the cylinder plus the
volume of the cone. I used the
maximum diameter of 18 feet, making
the radius a maximum of 9 feet.
1
𝜋𝑟 2 ℎ1 + 𝜋𝑟 2 ℎ2
3
1
𝜋92 ℎ1 + 𝜋92 ℎ2
3
I used the maximum total height of 25
feet. Since the volume of a cone
involves dividing by 3, I made the
height of the cone much smaller than
the height of the cylinder.
1
𝜋92 ℎ1 + 3 𝜋92 ℎ2
1
𝜋92 (23) + 𝜋92 (2) ≈ 6,022.43𝑓𝑡 3
3
Using ℎ1 = 23 𝑓𝑡 & ℎ2 = 2 𝑓𝑡, the silo
will have a volume greater than 5,886
cubic feet.
90. 446.2 cm3
91. 2598.78 ft3
92. 2.82 ft
93. 198.91 cm3
94. Sample Answer: Assuming that the
truck cars are rectangular prisms,
each truck holds 1,962 cubic feet of
grains (27.25 x 9 x 8 = 1,962). Two
truck cars hold 3,924 cubic feet of
grains. The volume of the silo equals
the volume of the cylinder plus the
volume of the cone. I used the
maximum diameter of 16 feet, making
the radius a maximum of 8 feet.
1
𝜋𝑟 2 ℎ1 + 𝜋𝑟 2 ℎ2
3
1
𝜋82 ℎ1 + 𝜋82 ℎ2
3
I used the maximum total height of 22
feet. Since the volume of a cone
involves dividing by 3, I made the
height of the cone much smaller than
the height of the cylinder.
Geometry – 3-D Geometry
~33~
1
𝜋82 ℎ1 + 3 𝜋82 ℎ2
1
𝜋82 (19) + 𝜋82 (3) ≈ 4,021.24𝑓𝑡 3
3
Using ℎ1 = 19 𝑓𝑡 & ℎ2 = 3 𝑓𝑡, the silo
will have a volume greater than 3,924
cubic feet.
95. 8000 mi
96. r = 7.2 u
97. r = 7.6 u
98. 314 yd2
99. 8.9"
100. a) 45238.9 m2
b) 1809.56 cm2
c) 53.79 mm2
101. 52 mm2
102. r = 3.99 cm
103. a) 907.9 m2
b) 615.75 cm2
104. a) 7 cm
b) 42 cm
c) 1,847.26 cm2
105. a) 288,000 m3
b) 457.3 cm3
366.16
2197
c) 𝜋2 mm3 = 6𝜋2 𝑚𝑚3
106. 265.96 cm3
107. a) 2,572.4 m3
b) 1,436.75 cm3
108. a) 808.17 cm3
b) 538.78 cm3
c) 66⅔%
109. 279,601 mi
110. 10.6 u
111. 5.2 u
112. 201.06 yd2
113. 2.86 in
114. a) 8,100 m2
b) 256 cm2
729
c) 𝜋 mm2
115. r = 1.67 cm
116. a) 615.75 m2
b) 572.56 m2
117. a) 4.5 cm
b) 27 cm
c) 763.4 cm2
118. a) 121,500m3
b) 682.6cm3
3,280.5
6561
c) 𝜋2 mm3 = 2𝜋2 𝑚𝑚3
119. 30.06 cm2
NJCTL.org
120. a) 1,436.76 m3
b) 1,288.2 m3
121. a) 5,176.56 cm3
b) 3,451.04 cm3
c) 66⅔%
122. a) 2.99
b) 2.26
123. Figure 1 and 2 because heights are
the same AND their cross sections
have equal area.
124. Sample Answer: The volume of each
congruent chunk is half the volume of
the cylinder. The volume of the
cylinder is 1029𝜋 𝑐𝑚3 ≈ 3232.70 cm3.
Therefore the volume of each
congruent chunk is 514.5𝜋 𝑐𝑚3 ≈
1,616.35 cm3. Each cube will have a
side length of 5 cm. The volume of
each clay cube will be 53 = 125 𝑐𝑚3 .
To find the number of cubes that Ann
can make from the chunk of clay,
divide the volume of one chunk of
clay by the volume of one cube.
1,616.35 ÷ 125 ≈ 12.93. The result
12.93 means that there is enough
clay in the chunk to make 12 clay
cubes because there is not enough
clay to make 13 complete cubes.
125. C and D
126. a) 26.18
b) 4
127. Figure1, 2, and 3 all have the same
heights but only figure 2 and 3 have
equal area cross sections.
128. A and E
129. Sample Answer: The volume of each
congruent chunk is half the volume of
the rectangular prism. The volume of
the prism is 360 𝑖𝑛3 . Therefore the
volume of each congruent chunk is
180 𝑖𝑛3 . Each cylinder will have a
radius of 1.5 inches and a height of 3
inches. The volume of each clay
cylinder will be 𝜋(1.5)2 (3) =
6.75𝜋 𝑖𝑛3 ≈ 21.21 𝑖𝑛3 . To find the
number of cylinders that Rick can
make from the chunk of clay, divide
the volume of the one chunk of clay
by the volume of one cylinder. 180 ÷
21.21 ≈ 8.49. The result 8.49 means
that there is enough clay in the chunk
to make 8 cylinders because there is
not enough clay to make 9 complete
cylinders.
130. a) Yes; k = 4 or ¼
b) No
131. a) 2 to 5
b) 4 to 25
c) 8 to 125
132. a) 4 to 3
b) 16 to 9
c) 64 to 27
133. a) 2 to 1
b) 4 to 1
c) 8 to 1
134. a) Yes; k = 2 or k = ½
b) No
135. a) 6 to 7
b) 36 to 49
c) 216 to 343
136. a) 5 to 2
b) 25 to 4
c) 125 to 8
137. a) 3 to 5
b) 9 to 25
c) 27 to 125
3-D Geometry: Review
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. 195.52 units3
11. A
12. B and F
13. SA = 192𝜋 𝑖𝑛2 ≈ 603.19 𝑖𝑛2
V = 384𝜋 𝑖𝑛3 ≈ 1,206.37 𝑖𝑛3
14. 147𝜋 𝑓𝑡 2 ≈ 461.81 𝑓𝑡 2
C
A
A
48 units2
2 cm
21𝜋 𝑢𝑛𝑖𝑡𝑠 2
208𝜋 𝑢𝑛𝑖𝑡𝑠 2 ≈ 653.45 𝑢𝑛𝑖𝑡𝑠 2
60 units2
96𝜋 𝑓𝑡 3 ≈ 301.59 𝑓𝑡 3
Geometry – 3-D Geometry
~34~
NJCTL.org
3-D Geometry: Review - Extended Constructed Response
1. A.
B. 416 cm2
C. 448 cm3
2. Sample Answer: Given that the salt water is transported in cylindrical containers, each
container holds 502.65 cubic feet of salt water
𝜋(4)2(10) = 160𝜋 = 502.65
Four containers will hold 640𝜋 = 2,010.62 cubic feet of salt water. The volume of the
new aquarium equals the volume of the prism plus the volume of the pyramid. I used
the maximum length of 16 feet and the maximum width of 12 feet.
16(12)h1 + (1/3)(16)(12)h2
I used the maximum total height of 18 feet. Since the volume of a pyramid involves
dividing by 3, I made the height of the pyramid much smaller than the height of the
prism.
16(12)h1 + (1/3)(16)(12)h2
16(12)(15) + (1/3)(16)(12)(3) = 3,072 cubic feet
Using h1 = 15 feet & h2 = 3 feet, the aquarium will have a volume greater than 2,010.62
cubic feet.
Note: any two heights that have a sum of 18 and create a volume greater than 2,010.62 are
acceptable.
Geometry – 3-D Geometry
~35~
NJCTL.org