3-d geometry intro to 3-d solids class...

Geometry – 3-D Geometry ~1~ NJCTL.org

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


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.


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.


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.


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.


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


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.


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.


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.


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.


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. c.


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.


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.


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.


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.


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.


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.


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.


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.


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?


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. Find the volume of the sphere. a. 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.


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.


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.


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.


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. 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?

132. 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?

133. 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. 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?

136. 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?

137. 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)?


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.


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.

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.

30 cm 30 cm


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?


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.


Answer Key 1.

a.)Edges:

𝐴𝐵̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ , 𝐷𝐸̅̅ ̅̅ , 𝐷𝐹̅̅ ̅̅ , 𝐴𝐹̅̅ ̅̅ , 𝑄𝑅̅̅ ̅̅ , 𝑅𝑆̅̅̅̅ , 𝑆𝑇̅̅̅̅ , 𝑇𝑈̅̅ ̅̅ , 𝑈𝑉̅̅ ̅̅ , 𝑉𝑄̅̅ ̅̅ , 𝐴𝑉̅̅ ̅̅ , 𝐵𝑄̅̅ ̅̅ , 𝐶𝑅̅̅ ̅̅ , 𝐷𝑆̅̅ ̅̅ , 𝐸𝑇̅̅̅̅ , 𝑈𝐹̅̅ ̅̅ ; Faces: ABCDEF, QRSTUV, CRSD, DSTE, ETUF, FUVC, ABQR; Vertices: A, B, C, D, E, F, Q, R, S, T, U, V

b.) Edges:

𝑉𝑍̅̅̅̅ , 𝑉𝑌̅̅ ̅̅ , 𝑉𝑋̅̅ ̅̅ , 𝑉𝑊̅̅ ̅̅ ̅, 𝑊𝑍̅̅ ̅̅ ̅, 𝑋𝑊̅̅ ̅̅ ̅, 𝑋𝑌̅̅ ̅̅ , 𝑌𝑍̅̅̅̅ ; Faces: VZY, VXY, VWX, VWZ, WXYZ; Vertices: V, W, X, Y, Z

c.) Edges:

𝑇𝑅̅̅ ̅̅ , 𝑇𝑆̅̅̅̅ , 𝑆𝑇̅̅̅̅ , 𝐽𝐾̅̅ ̅, 𝐽�̅�, 𝐾𝐿̅̅ ̅̅ , 𝑅𝐿̅̅̅̅ , 𝑆𝐾̅̅ ̅̅ , 𝑇𝐽̅̅ ̅; Faces: RST, JKL, TRLJ, TSKJ, RSKL; Vertices: J, K, L, R, S, T

2. a.) Base edges:

𝑃𝐴̅̅ ̅̅ , 𝐴𝑇̅̅ ̅̅ , 𝑇𝑁̅̅ ̅̅ , 𝑁𝐸̅̅ ̅̅ , 𝐸𝑃̅̅ ̅̅ , 𝐺𝐵̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ , 𝐷𝐹̅̅ ̅̅ , 𝐹𝐺̅̅ ̅̅ Lateral edges:

𝑃𝐺̅̅ ̅̅ , 𝐴𝐵̅̅ ̅̅ , 𝑇𝐶̅̅̅̅ , 𝑁𝐷̅̅ ̅̅ , 𝐸𝐹̅̅ ̅̅

b.) Base edges:

𝐴𝐵̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ , 𝐷𝐸̅̅ ̅̅ , 𝐸𝐹̅̅ ̅̅ , 𝐹𝐺̅̅ ̅̅ , 𝐺𝐴̅̅ ̅̅ Lateral edges:

𝑉𝐴̅̅ ̅̅ , 𝑉𝐵̅̅ ̅̅ , 𝑉𝐶̅̅̅̅ , 𝑉𝐷̅̅ ̅̅ , 𝑉𝐸̅̅ ̅̅ , 𝑉𝐹̅̅ ̅̅ , 𝑉𝐺̅̅ ̅̅

c.) Base edges:

𝑊𝑋̅̅ ̅̅ ̅, 𝑋𝑄̅̅ ̅̅ , 𝑄𝑀̅̅ ̅̅ ̅, 𝑀𝑊̅̅ ̅̅ ̅̅ , 𝑁𝑃̅̅ ̅̅ , 𝑃𝑌̅̅ ̅̅ , 𝑌𝑍̅̅̅̅ , 𝑍𝑁̅̅ ̅̅ Lateral edges:

𝑀𝑁̅̅ ̅̅ ̅, 𝑊𝑍̅̅ ̅̅ ̅, 𝑋𝑌̅̅ ̅̅ , 𝑄𝑃̅̅ ̅̅

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

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 = 𝜋𝑟2

48 = 𝑟2

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:

𝐴𝐵̅̅ ̅̅ , 𝑇𝐶̅̅̅̅ ,𝑁𝐷̅̅ ̅̅ , 𝐸𝐹̅̅ ̅̅ ,𝑃𝐺̅̅ ̅̅ , 𝐴𝑇̅̅ ̅̅ ,𝑇𝑁̅̅ ̅̅ , 𝑁𝐸̅̅ ̅̅ ,𝐸𝑃̅̅ ̅̅ , 𝑃𝐴̅̅ ̅̅ ,𝐵𝐶̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ ,𝐷𝐹̅̅ ̅̅ , 𝐹𝐺̅̅ ̅̅ ,𝐺𝐵̅̅ ̅̅ Faces: ATNEP, BCDFG, ABCT, TCDN, NDFE, EFGP, PGBA Vertices: A, B, C, D, E, F, G, N, P

b) Edges:

𝐴𝐵̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ , 𝐷𝐸̅̅ ̅̅ , 𝐸𝐺̅̅ ̅̅ , 𝐹𝐺̅̅ ̅̅ , 𝐺𝐴̅̅ ̅̅ , 𝐴𝑉̅̅ ̅̅ ,𝐵𝑉̅̅ ̅̅ , 𝐶𝑉̅̅ ̅̅ ,𝐷𝑉̅̅ ̅̅ , 𝐸𝑉̅̅ ̅̅ , 𝐹𝑉̅̅ ̅̅ , 𝐺𝑉̅̅ ̅̅ Faces: ABCDEFG, ABV, BCV, CDV, DEV, EFV, GFV, AGV Vertices: A, B, C, D, E, F, G, V

c) Edges:

𝑀𝑄̅̅ ̅̅ ̅, 𝑀𝑊̅̅ ̅̅ ̅̅ , 𝑊𝑋̅̅ ̅̅ ̅, 𝑋𝑄̅̅ ̅̅ , 𝑁𝑃̅̅ ̅̅ , 𝑃𝑌̅̅ ̅̅ , 𝑌𝑍̅̅̅̅ , 𝑍𝑁̅̅ ̅̅ , 𝑀𝑁̅̅ ̅̅ ̅, 𝑊𝑍̅̅ ̅̅ ̅, 𝑋𝑌̅̅ ̅̅ , 𝑄𝑃̅̅ ̅̅

Faces: QMWX, NPYZ, QXYP, MWZN Vertices: Q, P, M. N, W, X, Y, Z


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

9. 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

13. a

b

14. a b

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


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 = 5,616in2

27. LA = 252cm2

SA = 350 cm2 28. 416.75 m2 29. 33 ft2

30. LA = 1,319.47 cm2; used dh 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

49. 831.39 m2 50. 9.2 ft 51. 5 cm 52. 9.21 units

53. a) LA = 252 ft2 & SA = 396ft2 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) 1620m3

b) 450u3 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


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.

𝜋𝑟2ℎ1 +1

3𝜋𝑟2ℎ2

𝜋92ℎ1 +1

3𝜋92ℎ2

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.

𝜋92ℎ1 +1

3𝜋92ℎ2

𝜋92(23) +1

3𝜋92(2) ≈ 6,022.43𝑓𝑡3

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.

𝜋𝑟2ℎ1 +1

3𝜋𝑟2ℎ2

𝜋82ℎ1 +1

3𝜋82ℎ2

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.

𝜋82ℎ1 +1

3𝜋82ℎ2

𝜋82(19) +1

3𝜋82(3) ≈ 4,021.24𝑓𝑡3

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

c) 366.16

𝜋2 mm3 = 2197

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

c) 729

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) 121,500m3 b) 682.6cm3

c) 3,280.5

𝜋2 mm3 = 6561

2𝜋2 𝑚𝑚3

119. 30.06 cm2


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. C 2. A 3. A 4. 48 units2 5. 2 cm

6. 21𝜋 𝑢𝑛𝑖𝑡𝑠2

7. 208𝜋 𝑢𝑛𝑖𝑡𝑠2 ≈ 653.45 𝑢𝑛𝑖𝑡𝑠2 8. 60 units2

9. 96𝜋 𝑓𝑡3 ≈ 301.59 𝑓𝑡3

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 𝑓𝑡2 ≈ 461.81 𝑓𝑡2


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.

3-d geometry intro to 3-d solids class...

Documents