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Slide 1 / 138
3-D Geometry
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Table of Contents
Surface Area· Prisms· Pyramids· Cylinders
· Prisms and CylindersVolume
· Pyramids, Cones & Spheres
Nets
Click on the topic to go to that section
More Practice/ Review· Spheres
3-Dimensional Solids
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3-Dimensional Solids
Return toTable ofContents
Slide 4 / 138
The following link will take you to a site with interactive 3-D figures and nets.
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Polyhedron A 3-D figure whose faces are all polygons
Polyhedron Not Polyhedron
Sort the figures into the appropriate side.
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3-Dimensional SolidsCategories & Characteristics of 3-D 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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3-Dimensional SolidsCategories & Characteristics of 3-D 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
Slide 8 / 138
3-Dimensional Solids
Vocabulary Words for 3-D Solids:
Polyhedron A 3-D figure whose faces are all polygons (Prisms & Pyramids)
Face Flat surface of a Polyhedron
Edge Line segment formed where 2 faces meet
Vertex (Vertices) Point where 3 or more faces/edges meet
Slide 9 / 138Sort the figures.If you are incorrect, the figure will be sent back.
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Slide 15 / 138For 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 3-Dimensional Figures?
Name Faces Vertices Edges
Cube 6 8 12
Rectangular Prism
6 8 12
Triangular Prism
5 6 9
Triangular Pyramid
4 4 6
Square Pyramid
5 5 8
Pentagonal Pyramid
6 6 10
Octagonal Prism
10 16 24
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Euler's Formula
F + V - 2 = E
The number of edges is 2 less than the sum of the faces and vertices.
click to reveal
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Nets
Return toTable ofContents
Slide 21 / 138
NetsNets are two-dimensional drawings that represent the surface area of three-dimensional shapes.
There is more than one way to draw a net for a cube, however not all nets can be folded into a cube...
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Cut arrangements of 6 squares out of grid paper.
How many different arrangements can be folded into a cube?
Hint:You just saw one arrangement that works and one that does not.
see next page for answers...
Nets - Flat Patterns Activity Part 1
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click to reveal the cube flat patterns
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Cut an arrangement out of grid paper to create a rectangular prism that is not a cube .
Now make another flat pattern for the same rectangular prism.
Try each pattern out by folding it into a box.
What are the dimensions of each face of your rectangular prism?
Nets - Flat Patterns Activity Part 2
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Nets for prisms will have rectangular faces and two bases for which the shape is named.
Notice the two triangles areopposite from one another (bases).
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Volume
Return toTable ofContents
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Volume
Volume
- The amount of space occupied by a 3-D Figure - The number of cubic units needed to FILL a 3-D Figure (layering)
Label
Units3 or cubic units
click to reveal
click to reveal
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Volume of Prisms & Cylinders
Return toTable ofContents
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VolumeVolume of Prisms & Cylinders:
Area of Base x Height
Area Formulas:
Rectangle = lw or bh
Triangle = bh or 2 Circle = r2
click to reveal
(bh)
click to reveal
click to reveal
click to reveal
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Find the Volume.
5 m
8 m
2 m
Ans
wer
Ans
wer
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Find the Volume.
Ans
wer
Ans
wer
10 yd
9 yd
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14 Find the Volume.
7 in 1 5
1 in 1 2
4 in
Ans
wer
Ans
wer
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15 Find the volume of a rectangular prism with length 2 cm, width 3.3 cm and height 5.1 cm.
Ans
wer
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17 Find the volume.
21 ft
42 ft
50 ft47 ft A
nsw
er
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18 A box-shaped 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!
Ans
wer
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19 Find the volume.
6 m
10 m
Ans
wer
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20 Which circular glass holds more water?
A Glass A having a 7.5 cm diameter and standing 12 cm high
B Glass B having a 4 cm radius and a height of 11.5 cm
Ans
wer
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21 What is the volume of the largest cylinder that can be placed into a cube that measures 10 feet on an edge?
Ans
wer
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22 A circular garden has a diameter of 20 feet andis surrounded by a concrete border that has a width ofthree feet and a depth of 6 inches. What is the volume of concrete in the path? Use 3.14 for .#
Ans
wer
Slide 48 / 138
Volume of Pyramids, Cones & Spheres
Return toTable ofContents
Slide 49 / 138 Slide 50 / 138
Demonstration comparing volume of Cones & Spheres with volume of
Cylinders
click to go to web site
Slide 51 / 138
Volume of a Cone
(Area of Base x Height) 3
(Area of Base x Height) 1 3click to reveal
A cone is 1/3 the volume of a cylinder with the same base area (B) and height (h).
Slide 52 / 138
V = 2/3 (Volume of Cylinder)
r2 h( )2/3 V=or
V = 4/3 r3
Volume of a Sphere
click to reveal
A sphere is 2/3 the volume of a cylinder with the same base area (B) and height (h).
Slide 53 / 138
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?
(Just Ice Cream within Cone. Not on Top)Volume and Mass used in portion control. $$$
Pul
lP
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23 Find the volume.
4 in
9 inA
nsw
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24 Find the Volume
5 cm8 cm
Ans
wer
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25 What is the volume of a sphere with a radius of 8 ft?A
nsw
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26 What is the volume of a sphere with a diameter of 4.25 in?
Ans
wer
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Volume of a Pyramid
(Area of Base x Height) 3
(Area of Base x Height) 1 3click to reveal
A pyramid is 1/3 the volume of a prism with the same base area (B) and height (h).
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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 length = 4 m
V = Bh13
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27 Find the Volume of a triangular pyramid with base edges of 8 in, base height of 4 in and a pyramid height of 10 in.
8 in
10 in
4 in
Ans
wer
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28 Find the volume.
8 cm
7 cm
15.3 cm
Ans
wer
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Surface Area
Return toTable ofContents
Slide 64 / 138
Surface Area of Prisms
Return toTable ofContents
Slide 65 / 138
Surface AreaThe sum of the areas of all outside surfaces of a 3-D figure.
To find surface area, you must find the area of each surface of the figure then add them together.
6 in
3 in8 in
What type of figure is pictured?
How many surfaces are there?
How do you find the area of each surface?
Slide 66 / 138
Surface Area
6 in
3 in8 in
Bottom Top Left Right Front Back SUM 3 6 8 24 x 8 x 3 x 6 24 24 24 18 18 48 48 18 18 48 +48 180 in2
3 in8 in
6 in
8 in
6 in
3 in
6 in
3 in8 in
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Arrangement of Unit Cubes
Find all the possible ways that 24 unit cubes can be arranged into a rectangular prism. Give each arrangements dimensions and surface area. Also make a sketch.
Length Width Height Volume Surface Area
Sketch
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Arrangement of Unit Cubes
Find all the possible ways that 64 unit cubes can be arranged into a rectangular prism. Give each arrangements dimensions and surface area. Also make a sketch.
Length Width Height Volume Surface Area
Sketch
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29 Which arrangement of 27 cubes has the least surface area?
A 1 x 1 x 27
B 3 x 3 x 3
C 9 x 3 x 1
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30 Which arrangement of 12 cubes has the least surface area?
A 2 x 2 x 3
B 4 x 3 x 1
C 2 x 6 x 1
D 1 x 1 x 12
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31 Which arrangement of 25 cubes has the greatest surface area?
A 1 x 1 x 25
B 1 x 5 x 5
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32 Which arrangement of 48 cubes has the least surface area?
A 4 x 12 x 1
B 2 x 2 x 12
C 1 x 1 x 48
D 3 x 8 x 2
E 1 x 2 x 24
F 4 x 3 x 4
G 1 x 8 x 6
H 4 x 2 x 6
I 3 x 16 x 1
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Find the surface area of a rectangular shoe box that has a length of 12 inches, a width of 6 inches and a height of 5 inches.
Top/Bottom Front/Back Left/Right Surface Area 12 12 6 144 x 6 x 5 x 5 120 72 60 30 + 60 x 2 x 2 x 2 324 in2 144 120 60
5 in
6 in
12 in
click to reveal click to reveal click to reveal click to reveal
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Find the Surface Area.1. Draw and label ALL faces2. Find the correct dimensions for each face3. Calculate the AREA of EACH face4. Find the SUM of ALL faces5. Label Answer
5 yd
6 yd
4 yd
9 yd
5 yd
go on to see steps
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Triangles Bottom Rectangle 4 9 x 6 x 6 24 / 2 = 12 54 x 2 24 Total
SurfaceArea 24 54+ 90 168 yd2
5 yd
6 yd
4 yd9 yd
5 yd
9 yd
5 yd
5 yd
6 yd
4 yd
Left/Right Rectangles(Same size since isosceles) 5 x 9 45 x 2 90
9 yd
5 yd
5 yd
6 yd
4 ydClick on box to
reveal steps.
Click on box to reveal steps.
Click on box to reveal steps.
Click on box to reveal steps.
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Find the Surface Area.1. Draw and label ALL faces2. Find the correct dimensions for each face3. Calculate the AREA of EACH face4. Find the SUM of ALL faces5. Label Answer
9 cm
6 cm 11 cm
9 cm
9 cm
go on to see steps
Slide 86 / 138
Triangles Rectangles 9 (All 3 are the same size X6 since equilateral) 54 / 2 = 27 9 x 2 x 11 54 99 x 3 297
Surface Area 297+ 54 351 cm2
9 cm
6 cm 11 cm
9 cm
9 cm
9 cm
6 cm 11 cm
9 cm
9 cm
Click on box to reveal steps.
Click on box to reveal steps.
Slide 87 / 138
39 Find the surface area of the shape below.
21 ft
42 ft
50 ft
47 ft
1. Draw and label ALL faces2. Find the correct dimensions for each face3. Calculate the AREA of EACH face4. Find the SUM of ALL faces5. Label Answer
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7 cm 3 cm
4 cm
15 cm
6 cm
5 cm
Find the Surface Area.
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Trapezoids 10 + 6 16 x 4 64 / 2 = 32 x 2 64
Bottom Rectangle 6 x 15 90
Top Rectangle 10 x 15 150
Side Rectangles 5 x 15 75 x 2 150
Surface Area 64 90 150 + 150 454 cm2
7 cm 3 cm
4 cm
15 cm
6 cm
5 cm
click to reveal
click to reveal
click to reveal
click to reveal
click to reveal
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40 Find the surface area of the shape below.
8 cm 6 cm
10 cm
9 cm
Pul
lP
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41 Find the surface area of the shape below.
10 cm2 cm
6 cm
10 cm
6 cm
Pul
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Surface Area of Pyramids
Return toTable ofContents
Slide 93 / 138
Surface Area of Pyramids
What is a pyramid?
Polyhedron with one base and triangular faces that meet at a vertex
How do you find Surface Area?
Sum of the areas of all the surfaces of a 3-D Figureclick to reveal
click to reveal
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8 cm
8 cm
17.5 cm
Find the Surface Area.
go on to see steps
Slide 95 / 138
Find the Surface Area.
Bottom Rectangle 8 x 7 56
Front/Back Triangles
Left/RightTriangles
Surface Area 56 140+ 140 336 cm2
8 cm
8 cm
17.5 cm
8 cm
17.5 cm
8 cm
8 cm
17.5 cm
8 cm
Slide 96 / 138
Find the surface area of a square pyramid with base edge of 4 inches and triangle height of 3 inches.
4 in
3 inBase 4x 4 16
Triangles Surface Area 16+ 24 40 in2
click to reveal click to reveal click to reveal
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Find the surface area.
Be sure to look at the base to see if it is an equilateral or isosceles triangle (making all or two of the side triangles equivalent!).
Base Triangles(all equal)
Surface Area 7+ 36 43 in2
4 in4 in
4 in
6 in
3.5 in
click to reveal click to reveal
click to reveal
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42 Which has a greater Surface Area, a square pyramid with a base edge of 8 in and a height of 4 in or a cube with an edge of 5 in?
A Square Pyramid
B Cube
Pul
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43 Find the Surface Area of a triangular pyramid with base edges of 8 in, base height of 4 in and a slant height of 10 in.
8 in8 in
8 in
10 in
6.9 in
Pul
lP
ull
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44 Find the Surface Area.
9 m9 m
12 m
11 m
6.7 m
Pul
lP
ull
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Surface Area of Cylinders
Return toTable ofContents
Slide 102 / 138
How would you find the surface area of a cylinder?
Slide 103 / 138Notice the length of the rectangle is actually the circumference of the circular base.
Steps1. Find the area of the 2 circular bases.2. Find the area of the curved surface (actually, a rectangle).3. Add the two areas.4. Label answer.
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Radius
Curved Side = Circumference of Circular Base
HEIGHT
HEIGHT
Radius
Slide 105 / 138
Area of Circles = 2 ( )Area of Curved Surface = Circumference Height d Height
2 r2 + dh
2 r2 + 2 rh-or-
r2
Radius
Curved Side = Circumference of Circular Base
HEIGHT
HEIGHT
Radius
Slide 106 / 138
Find the surface area of a cylinder whose height is 14 inches and whose base has a diameter of 16 inches.
14 in
16 in Area of Circles = 2 ( r2 ) = 2 ( 82) = 2 (64 ) = 128 = 401.92 in2
Area of Curved Surface = Circumference Height = d Height = (16)(14) = 224 = 703.36 in2
Surface Area = 401.92 + 703.36 = 1105.28 in2
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45 Find the surface area of a cylinder whose height is 8 inches and whose base has a diameter of 6 inches.
Pul
lP
ull
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46 Find the surface area of a cylinder whose height is 14 inches and whose base has a diameter of 16 inches.
Pul
lP
ull
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47 How much material is needed to make a cylindrical orange juice can that is 15 cm high and has a diameter of 10 cm?
Pul
lP
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48 Find the surface area of a cylinder with a height of 14 inches and a base radius of 8 inches.
Pul
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ull
Slide 111 / 138
49 A cylindrical feed tank on a farm needs to be painted. The tank has a diameter 7.5 feet and a height of 11 ft. One gallon of paint covers 325 square feet. Do you have enough paint? Explain.
Yes
No
Pul
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Slide 112 / 138
Return toTable ofContents
Surface Area of Spheres
Slide 113 / 138
A sphere is the set of all points that are the same distance from the center point.
Like a circle, a sphere has a radius and a diameter.
You will see that like a circle, the formula for surface area of a sphere also includes pi.
Radius
Surface Area of a Sphereclick to reveal
Slide 114 / 138
If the diameter of the Earth is 12,742 km, what is its surface area?
12,742 km
Slide 115 / 138
Try This:
Find the surface area of a tennis ball whose diameter is 2.7 inches.
2.7 in
click to reveal
Slide 116 / 138
50 Find the surface area of a softball with a diameter 3.8 inches.
Pul
lP
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51 How much leather is needed to make a basketball with a radius of 4.7 inches?
Pul
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52 How much rubber is needed to make 6 racquetballs with a diameter of 5.7 inches?
Pul
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Slide 119 / 138
More Practice / Review
Return toTable ofContents
Slide 120 / 138
53 Find the volume.
15 mm
8 mm
22 mm
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54 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!
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55 Find the volume of a square pyramid with base edge of 4 inches and pyramid height of 3 inches.
Slide 123 / 138
56 Find the Volume
9 m9 m
12 m
11 m
6 m
Slide 124 / 138
57 Find the Volume
14 ft
21 ft
Slide 125 / 138
58 Find the Volume
8 in
6.9 in
Slide 126 / 138
59 Find the Volume
4 ft
7 ft
8 ft
9 ft
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60 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 cones-ful of soil were needed to fill the planter?
20 cm
14 cm
25 cm
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61 Find the surface area of the cylinder.
Radius = 6 cm and Height = 7 cm
Slide 129 / 138
62 Find the Surface Area.
11 cm
11 cm
12 cm
Slide 130 / 138
63 Find the Surface Area.
7 in8 in
9 in
9 in
2 in
Slide 131 / 138
64 Find the Volume.
7 in8 in
9 in
9 in
2 in
Slide 132 / 138
65 A rectangular storage box is 12 in wide, 15 in long and 9 in high. How many square inches of colored paper are needed to cover the surface area of the box?
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66 Find the surface area of a square pyramid with a base length of 4 inches and slant height of 5 inches.
Slide 134 / 138
Name a 3-D Figure that is not a polyhedron.
Slide 135 / 138
Define the following terms:
Surface Area
Volume
Slide 136 / 138
Name a 3-D figure that has 6 rectangular faces.
Slide 137 / 138
67 Find the volume.
40 m
70 m
80 m
Slide 138 / 138
68 A teacher made 2 pair of foam dice to use in math games. Each cube measured 10 in on each side. How many square inches of fabric were needed to cover the 2 cubes?