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 p.
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 p .
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 p .