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8th Grade3D Geometry
20151120
www.njctl.org
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Table of Contents
• Prisms and CylindersVolume
• Pyramids, Cones & Spheres
Click on the topic to go to that section
More Practice/ Review
3Dimensional Solids
Glossary & Standards
Teacher N
otes Vocabulary Words are bolded
in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it.
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3Dimensional Solids
Return toTable ofContents
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The following link will take you to a site with interactive 3D figures and nets.
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PolyhedronA 3D figure whose faces are all polygons.
Polyhedron Not Polyhedron
Sort the figures into the appropriate side.
Polyhedron
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3Dimensional SolidsCategories & Characteristics of 3D Solids:
Prisms1. Have 2 congruent, polygon bases which are parallel to one another2. Sides are rectangular (parallelograms)3. Named by the shape of their base
Pyramids1. Have 1 polygon base with a vertex opposite it2. Sides are triangular3. Named by the shape of their base
click to reveal
click to reveal
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3Dimensional SolidsCategories & Characteristics of 3D Solids:
Cylinders1. Have 2 congruent, circular bases which are parallel to one another2. Sides are curved
Cones1. Have 1 circular bases with a vertex opposite it2. Sides are curvedclick to reveal
click to reveal
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Edge Line segment formed where 2 faces meet
Vertex (Vertices)Point where 3 or more faces/edges meet
3Dimensional SolidsVocabulary Words for 3D Solids:
Polyhedron A 3D figure whose faces are all polygons (Prisms & Pyramids)
Face Flat surface of a Polyhedron
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Sort the figures.If you are incorrect, the figure will be sent back.
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1 Name the figure.
A Rectangular Prism B Triangular Pyramid C Hexagonal Prism D Rectangular PyramidE Cylinder F Cone
Answer
D
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2 Name the figure.
A Rectangular Pyramid B Triangular Prism C Octagonal Prism D Circular Pyramid E Cylinder F Cone
Answer
E
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3 Name the figure.
A Rectangular Pyramid B Triangular Pyramid C Triangular Prism D Hexagonal Pyramid E Cylinder F Cone
Answer
B
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4 Name the figure.
A Rectangular Prism B Triangular Prism C Square Prism D Rectangular Pyramid E Cylinder F Cone
Answer
A
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5 Name the figure.
A Rectangular Prism B Triangular Pyramid C Circular Prism D Circular Pyramid E Cylinder F Cone
Answer
F
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For each figure, find the number of faces, vertices and edges.Can you figure out a relationship between the number of faces, vertices and edges of 3Dimensional Figures?
Name Faces Vertices Edges Cube
Rectangular Prism Triangular PrismTriangular PyramidSquare PyramidPentagonal PyramidOctagonal Prism
Math Practice
MP.7: Look for and make use of structure.
MP.8: Look for and express regularity in repeated reasoning.
Ask: What patterns do you see in the numbers that represent the vertices,
edges, and faces?
What generalizations can you make about the number of vertices, edges,
and faces in a solid?
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Euler's Formula
F + V = E + 2
Euler's Formula is the number of edges plus 2 is equal to the sum of the faces and vertices.
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6 How many faces does a pentagonal prism have?
Answer
7
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7 How many edges does a rectangular pyramid have?
Answer
8
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8 How many vertices does a triangular prism have?
Answer
6
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9 How many faces does a hexagonal pyramid have?
Answer
7
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10 How many vertices does a triangular pyramid have?
Answer
4
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Volume
Return toTable ofContents
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Label Units3 or cubic units
Volume
Volume
The amount of space occupied by a 3D Figure The number of cubic units needed to FILL a 3D Figure (layering)click to reveal
click to reveal
Math Practice
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Volume Activity
Click the link below for the activity.
Lab #1: Volume Activity
Teacher N
otes
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Volume of Prisms & Cylinders
Return toTable ofContents
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Volume
Volume of Prisms & Cylinders:
Area of Base x Height, or V = Bh
Area Formulas:
Rectangle = lw or bh
Triangle = bh or 2 Circle = πr2
click to reveal
(bh) 1 2
click to reveal
click to reveal
click to reveal
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Find the Volume
5 m
8 m
2 m
Answer
VOLUME: 2x 5 10 (Area of Base)x 8 (Height)80 m3
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Find the Volume Use 3.14 as your value of π.
10 yd
9 yd
Answer
VOLUME: 9 x 9 81x 3.14 254.34 x 10 2543.4 yd3
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A cylinder with a radius measuring 2 cm and a height of 5 cm is compared to a cylinder with a radius of 4 cm and a height of 5 cm. Amy says that the volume of the cylinder with a radius of 4 cm is double the volume of the cylinder with a radius of 2 cm. She used 3.14 as her value of π. Is she correct? Explain your reasoning.
Start by calculating the volume of both cylinders.
V = (3.14)(2)2(5)
V = 62.8 cm3
V = (3.14)(4)2(5)
V = 251.2 cm3
Answer the question.
No, Amy is not correct. If the radius of the cylinder doubles, the volume does not double. Instead it quadruples.
click click
click
Find the Volume
Math Practice
MP.3 Construct viable arguments & critique the reasoning of others.After calculating the volume of both
cylinders, ask:
What do you think about what Amy predicted?
Do you agree? Why or Why not?
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Teachers:
Use this Mathematical Practice Pull Tab for the next 9 SMART Response slides.
Math Practice
MP.5 Use appropriate tools strategically.
Ask: Can you make a model to show that?
Would it help to create a diagram/draw a picture?
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4 in
11 Find the Volume.
7 in 1 5
1 in 1 2 A
nswer
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12 Find the volume of a rectangular prism with length 2 cm, width 3.3 cm and height 5.1 cm.
Answer
34
13 Which is a possible length, width and height for a rectangular prism whose volume = 18 cm 3
A 1 x 2 x 18B 6 x 3 x 3C 2 x 3 x 3D 3 x 3 x 3
Answer
C
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14 Find the volume.
21 ft
42 ft
50 ft47 ft
Answer
V = Bh & B = bh of the triangle
V = (21)(42)(50)
V = (882)(50)
V = 441(50)V = 22,050 ft3
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15 A boxshaped refrigerator measures 12 by 10 by 7 on the outside. All six sides of the refrigerator are 1 unit thick. What is the inside volume of the refrigerator in cubic units?
HINT: You may want to draw a picture!
Answer
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16 Find the volume. Use 3.14 as your value of π.
6 m
10 m
Answer
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17 Which circular glass holds more water?
Note: Use 3.14 as your value of π.
AGlass A having a 7.5 cm diameter and standing 12 cm high
BGlass B having a 4 cm radius and a height of 11.5 cm
Answer
Glass Ad = 7.5, so r = 3.75V = B hV = r2 hV = 3.14 (3.75)2 12V = 3.14 14.0625 12V = 529.875 cm3
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18 What is the volume of the largest cylinder that can be placed into a cube that measures 10 feet on an edge? Use 3.14 as your value of π.
Answer d = 10 ft, so r = 5 ft & h = 10 ft
V = π (52)(10)V = π(25)(10)V = 785 ft3
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19 A circular garden has a diameter of 20 feet andis surrounded by a concrete border that has a width of three feet and a depth of 6 inches. What is the volume of concrete in the path? Use 3.14 as your value of π.
Answer
41
Sometimes, a question will ask you to "Leave your answer in terms of π". This means that you treat π like a variable & only do the arithmetic operations with the remaining numbers.
Ex: If a cylinder has a radius of 3 and a height of 4, then
Volume = π(3)2(4)
= π(9)(4)
= 36π units2
Let's try some more problems like this one.
Click here to return to cones & spheres.
Answer in Terms of π
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Leave your answer in terms of π.
10 yd
9 yd
Find the Volume
Answer
43
30 ft
15 ft
Leave your answer in terms of π.
Find the Volume
Answer
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20 A cylinder has a radius of 7 and a height of 2. What is its volume? Leave your answer in terms of π.
A 14π units3
B 28π units3
C 49π units3
D 98π units3
Answer
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21 A cylinder has a diameter of 12 in. and a height of 12 in. What is its volume? Leave your answer in terms of π.
A 144π in3
B 432π in3
C 864π in3
D 1,728π in3
Answer
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22 A cylinder has a diameter of 17 in. and a height of 5 in. What is its volume? Leave your answer in terms of π.
A 106.25π in3
B 361.25π in3
C 425π in3
D 1,228.25π in3
Answer
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23 A circular pool has a diameter of 40 feet and is surrounded by a wooden deck that has a width of 4 feet and a depth of 6 inches. What is the volume of the wooden deck? Leave your answer in terms of π.
A 88π ft3
B 176π ft3
C 400π ft3
D 576π ft3
Answer
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Volume of Pyramids, Cones & Spheres
Return toTable ofContents
49
Given the same diameter and height for each figure, drag them to arrange in order of smallest to largest volume.
How many filled cones do you think it would take to fill the cylinder?
How many filled spheres do you think it would take to fill the cylinder?
Volume
Math Practice
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Demonstration comparing volume of Cones & Spheres with volume of
Cylinders
click to go to web site
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(Area of Base x Height) = Bh 1 3
1 3
A cone is 1/3 the volume of a cylinder with the same base area (B) and height (h).
Area of Base x Height3
Bh3=
Volume of a Cone
click to reveal
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V = 2/3 (Volume of Cylinder)
r2 h( )2/3 V=or
V = 4/3 r3ππ
Volume of a Sphere
A sphere is 2/3 the volume of a cylinder with the same base area (B) and height (h).
click to reveal
Figu
re
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How much ice cream can a Friendly’s Waffle cone hold if it has a diameter of 6 in and its height is 10 in? Use 3.14 as your value of π.
(Just Ice Cream within Cone. Not on Top)Volume and Mass used in portion control. $$$
Volume
Answer &
Math Practice
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24 Find the volume. Use 3.14 as your value of π.
4 in
9 in Answer
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25 Find the Volume. Use 3.14 as your value of π.
5 cm8 cm A
nswer
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V = πr3
V = (3.14)(5.5)3
V = 696.6 cm3
4 3 4 3
If the radius of a sphere is 5.5 cm, what is its volume? Use 3.14 as your value of π.
Click here
Volume
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26 What is the volume of a sphere with a radius of 8 ft? Use 3.14 as your value of π.
Answer
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27 What is the volume of a sphere with a diameter of 4.25 in? Use 3.14 as your value of π.
Answer
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Similar to when we found the volume of a cylinder, with a cone and a sphere, you could be asked to "Leave your answer in terms of π".
Click here if you need to review that property.
Volume in Terms of π
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You are selling lemonade in conic cups (cups shaped like cones). How much lemonade will each customer get to drink? Leave your answer in terms of π.
8 cm
11 cm
Volume in Terms of π
Answer
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If the radius of a sphere is 6 cm, what is its volume? Leave your answer in terms of π.
Volume in Terms of π
Answer
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28 Find the volume of the cone below. Leave your answer in terms of π.
A 12π in3
B 36π in3
C 48π in3
D 144π in3 4 in
9 in
Answer
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29 Find the volume of the sphere that has a diameter of 18 cm. Leave your answer in terms of π.
A 729π cm3
B 972π cm3
C 5,832π cm3
D 7,776π cm3
Answer
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30 Find the volume of the cone below. Leave your answer in terms of π.
A 49π in3
B 84π in3
C 147π in3
D 252π in3 7 in
12 in
Answer
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31 Find the volume of a sphere that has a radius of 4.5 cm. Leave your answer in terms of π.
A 27π cm3
B 91.125π cm3
C 121.5π cm3
D 364.5π cm3
Answer
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32 A sphere with a radius measuring 9 cm is compared to a sphere with a radius of 18 cm. Jeff says that the volume of the sphere with a radius of 18 cm is double the volume of the sphere with a radius of 9 cm. Is he correct? Explain your reasoning. When you are done calculating your answer, type in the number "1".
Answer
V = (9)3
V = (729)
V =
V = 972 cm3
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(Area of Base x Height) = Bh 1 3
1 3
Area of Base x Height3
Bh3=
Volume of a Pyramid
A pyramid is 1/3 the volume of a prism with the same base area (B) and height (h).
click to reveal
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Pyramids are named by the shape of their base..
The volume is a pyramid is 1/3 the volume of a prism with the same base area(B) and height (h).
V = Bh13
=5 m
side le
ngth = 4 m
V = Bh
V = (4)(4)(5)
V = (80)
V = 26 m3
1 3 1 3 1 3
2 3
Click here
Pyramids
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33 Find the Volume of a triangular pyramid with a base edge of 8 in, base height of 4 in and a pyramid height of 10 in.
8 in
10 in
4 in
Answer
70
34 Find the volume.
8 cm
7 cm
15.3 cm
Answer
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More Practice / Review
Return toTable ofContents
72
35 Find the volume.
15 mm
8 mm
22 mm
Answer
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36 Find the volume of a rectangular pyramid with a base length of 2.7 meters and a base width of 1.3 meters, and the height of the pyramid is 2.4 meters.
HINT: Drawing a diagram will help!
Answer
74
37 Find the volume of a square pyramid with base edge of 4 inches and pyramid height of 3 inches.
Answer
75
38 Find the Volume.
9 m9 m
12 m
11 m
6 m
Answer
76
39 Find the Volume. Use 3.14 as your value of π.
14 ft
21 ft
Answer
77
40 Find the Volume. Use 3.14 as your value of π.
8 in
6.9 in
Answer
78
41 Find the Volume.
4 ft
7 ft
8 ft
9 ft
Answer
79
42 A cone 20 cm in diameter and 14 cm high was used to fill a cubical planter, 25 cm per edge, with soil. How many full cones of soil were needed to fill the planter?
20 cm
14 cm
25 cm
Answer
Cone Cube1/3(3.14)(102)(14) 2531465.3 cm3 15625 cm3
15625/1465.3 ≈ 10.7 about 11 cones
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43 Find the Volume.
7 in8 in
9 in
9 in
2 in
Answer
81
Name a 3D Figure that is not a polyhedron.
Answer
82
Name a 3D figure that has 6 rectangular faces.
Answer
83
44 Find the volume.
40 m
70 m
80 m
Answer
84
45 The figure shows a right circular cylinder and a right circular cone. The cylinder and the cone have the same base and the same height.
Part A: What is the volume of the cone, in cubic feet?
From PARCC EOY sample test calculator #11
A 12π ft3
B 16π ft3
C 36π ft3
D 48π ft3
Answer
85
46 Part B: What is the ratio of the cone's volume to the cylinder's volume?
Answer
86
Glossary & Standards
Return toTable ofContents
Teacher N
otes Vocabulary Words are bolded
in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it.
87
tip
traffic cone
cone
pencil
ice cream
Back to Instruction
curved
polyhedron
surface
Cone A polyhedron that has one circular base with a vertex opposite of it
and sides that are curved.
88
candles
pizza
Pringlescan
Back to Instruction
curved
polyhedron
surface
Cylinder A polyhedron that has two congruent circular bases which are parallel to one another and sides that are curved.
89
Back to Instruction
Edge
Line segment formed where 2 faces meet.
A triangular pyramid has 6
edges.
90
Back to Instruction
Euler's Formula
The number of edges plus 2 is equal to the sum of the faces and vertices.
E + 2 = F + V
E + 2= F + VE + 2 = 4 + 4E + 2 = 8E = 6faces = 4
vertices = 4pyramid:
91
Back to Instruction
Face
Flat surface of a polyhedron.
A triangular pyramid has 4 faces. (there is
one you can't see)
92
Back to Instruction
Polyhedron
A 3D figure whosefaces are all polygons.
CubesPrismsPyramids
Made of:
Faces
Edges
Vertices
Cylinders
Cones
93
RectangularPrism
Triangular
Prism
Prism
Pentagonal
Back to Instruction
Prism A polyhedron that has two congruent, polygon bases which are parallel to one another, sides that are rectangular,
and named by the shape of their base.
Block of cheese
bodyof
pencil
juicebox
94
Pentagonal Pyramid
SquarePyramidTriangular
Pyramid
Back to Instruction
Pyramid
A polyhedron that has one polygon base with a vertex opposite of it,sides that are triangular,
and named by the shape of its base.
95
Back to Instruction
Vertex
Point where two or more straight lines/faces/edges meet.
A Corner.
A triangular pyramid has 4
vertices.
96
Back to Instruction
Volume
The number of cubic units needed to fill a 3D figure (layering).
The amount of space occupied by a 3D figure.
Label: Units3 or
cubic units
volume of prisms
and cylinders:
area of base x height
V = area of base x h V = 2m x 5m x 8m V = 80m3
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Standards for Mathematical Practices
Click on each standard to bring you to an example of how to meet this standard within the unit.
MP8 Look for and express regularity in repeated reasoning.
MP1 Make sense of problems and persevere in solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique the reasoning of others.
MP4 Model with mathematics.
MP5 Use appropriate tools strategically.
MP6 Attend to precision.
MP7 Look for and make use of structure.
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