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New Jersey Center for Teaching and Learning
7th Grade
Progressive Mathematics Initiative
This material is made freely available at www.njctl.org
and is intended for the non-commercial use of
students and teachers. These materials may not be
used for any commercial purpose without the written
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community, and/or provide access to course
materials to parents, students and others.
3-D Geometry
2013-02-22
www.njctl.org
Click to go to website:
www.njctl.org
Slide 3 / 135
Table of Contents
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Click on the topic to
go to that section
3-Dimensional Solids
Cross Sections of 3-Dimensional Figures
Volume
· Prisms and Cylinders
· Pyramids, Cones & Spheres
3-Dimensional Solids
Surface Area
·
·
·
·
Prisms
Pyramids
Cylinders
Spheres
Return to
Table of
Contents
More Practice/ Review
Common Core: 7.G.3, 7.G.6, 7.EE.3
Slide 5 / 135
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Polyhedron A 3-D figure whose faces are all polygons
Sort the figures into the appropriate side.
The following link will take you to a site with
interactive 3-D figures and nets.
Polyhedron
Not Polyhedron
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3-Dimensional Solids
3-Dimensional Solids
Categories & Characteristics of 3-D Solids:
Categories & Characteristics of 3-D Solids:
Prisms
1. Have 2 congruent, polygon bases which are parallel
to one another
to reveal
2. Sides are rectangularclick
(parallelograms)
3. Named by the shape of their base
Cylinders
1. Have 2 congruent, circular bases which
are parallel to oneclick
another
to reveal
2. Sides are curved
Pyramids
1. Have 1 polygon base with a vertex opposite it
2. Sides are triangular
click to reveal
3. Named by the shape of their base
Cones
1. Have 1 circular bases with a vertex opposite it
2. Sides are curved
click to reveal
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Slide 10 / 135
Sort the figures.
If you are incorrect, the figure will be sent back.
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 11 / 135
1
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2
Name the figure.
Name the figure
A
Rectangular Prism
A
Rectangular Pyramid
B
Triangular Pyramid
B
Triangular Prism
C
Octagonal Prism
C
Hexagonal Prism
D
Rectangular Pyramid
E
Cylinder
F
Cone
Pull
D
Circular Pyramid
E
Cylinder
F
Cone
Pull
Slide 13 / 135
3
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Name the figure
4
A
Rectangular Pyramid
B
Triangular Pyramid
C
Triangular Prism
D
Hexagonal Pyramid
E
Cylinder
F
Cone
Pull
Name the figure
A
Rectangular Prism
B
Triangular Prism
C
Square Prism
D
Rectangular Pyramid
E
Cylinder
F
Cone
Slide 15 / 135
5
Slide 16 / 135
For each figure, find the number of faces, vertices and edges.
Name the figure
Can you figure out a relationship between the number of faces,
vertices and edges of 3-Dimensional Figures?
A
Rectangular Prism
B
Triangular Pyramid
C
Circular Prism
D
Circular Pyramid
E
Cylinder
F
Cone
Pull
Vertices
Edges
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
Name
Pull
Slide 17 / 135
Euler's Formula
Faces
Cube
Slide 18 / 135
6
How many faces does a pentagonal prism have?
F+V-2=E
click to reveal
The number of edges is 2 less than
the sum of the faces and vertices.
Pull
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7
Slide 20 / 135
How many edges does a rectangular pyramid have?
8
How many vertices does a triangular prism have?
Pull
Pull
Slide 21 / 135
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3-Dimensional figures can be cut by planes. When you cut
a 3-D figure by a plane, the result is a 2-D figure.
Cross Sections of
Three-Dimensional
Figures
The cross-sections of 3-D figures are 2 dimensional figures
you are familiar with.
Look at the example on the next page to help your
understanding.
Return to
Table of
Contents
Slide 23 / 135
A horizontal cross-section of a cone is a circle.
Can you describe a vertical cross-section of a cone?
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A vertical cross-section of a cone is a triangle.
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The horizontal cross-section is a circle.
The vertical cross-section is a rectangle
A water tower is built in the shape of a cylinder.
How does the horizontal cross-section compare to the vertical
cross-section?
Slide 27 / 135
10 Which figure does not have a triangle as one of its cross-
Which figure has the same horizontal and vertical crosssections?
C
B
D
A
C
B
D
Pull
A
sections?
Pull
9
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11 Which is the vertical cross-section of the figure shown?
A Triangle
A Triangle
B Circle
B Circle
C
Square
C
Square
D
Trapezoid
D
Trapezoid
Pull
Pull
12 Which is the horizontal cross-section of the figure shown?
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13 Which is the vertical cross-section of the figure shown?
Pull
Volume
A Triangle
Return to
Table of
Contents
B Circle
C
Square
D
Trapezoid
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Volume
Volume Activity
Volume
- The amount of space occupied by a 3-D Figure
click
to reveal
- The number of cubic units
needed
to FILL a 3-D Figure (layering)
Label
Take unit cubes and create a rectangular prism with
dimensions of 4 x 2 x 1.
What happens to the volume if you add another layer and
make it 4 x 2 x 2?
What happens to the volume if you add another layer and
make it 4 x 2 x 3?
3
click
reveal
Units
orto
cubic
units
for teacher note
Pull
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Volume
Volume of Prisms & Cylinders:
Area of
Base
x Height
click
to reveal
Volume of Prisms & Cylinders
Area Formulas:
click
to reveal
Rectangle = lw
or bh
Return to
Table of
Contents
Triangle = click
bh or
to reveal
(bh)
2
Circle = click
r2 to reveal
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Find the Volume.
Answer
Answer
Find the Volume.
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9 yd
8m
5m
Answer
Answer
2m
10 yd
Slide 39 / 135
Find the Volume.
15
Find the volume of a rectangular prism with length 2 cm,
width 3.3 cm and height 5.1 cm.
Answer
Answer
14
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4 in
1 12 in
Answer
in
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16
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Which is a possible length, width and height for a
rectangular prism whose volume = 18 cm 3
A
1 x 2 x 18
B
6x3x3
C
2x3x3
D
3x3x3
17
Find the volume.
Answer
7
1
5
50 ft
Pull
47 ft
42 ft
21 ft
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19
6m
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21
20 Which circular glass holds more water?
B
Glass B having a 4 cm radius and a height
of 11.5 cm
What is the volume of the largest cylinder that can be
placed into a cube that measures 10 feet on an edge?
Answer
Glass A having a 7.5 cm diameter and
standing 12 cm high
Answer
10 m
HINT: You may want to draw a picture!
A
Find the volume.
22
A circular garden has a diameter of 20 feet and
is 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 for π .
Slide 48 / 135
Answer
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Volume of Pyramids, Cones & Spheres
Return to
Table of
Contents
Answer
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?
Answer
18
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Demonstration comparing volume of
Cones & Spheres with volume of
Cylinders
click to go to web site
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Volume of a Sphere
Volume of a Cone
A cone is 1/3 the
volume of a cylinder
with the same base area
(B) and height (h).
(Area of Base x Height)
1
3
3
A sphere is 2/3 the
volume of a cylinder
with the same base area
(B) and height (h).
(Area of Base x Height)
V = 2/3 (Volume of Cylinder)
2/3 (
clickV=
to reveal
or
V = 4/3
r2 h )
r3
click to reveal
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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?
23
Find the volume.
Pull
Answer
(Just Ice Cream within Cone. Not on Top)
9 in
4 in
Slide 55 / 135
Find the Volume
If the radius of a sphere is 5.5 cm, what is its volume?
5 cm
Slide 57 / 135
What is the volume of a sphere with a radius of 8 ft?
26
What is the volume of a sphere with a diameter of 4.25 in?
Answer
25
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Volume of a Pyramid
A pyramid is 1/3 the
volume of a prism with
the same base area (B)
and height (h).
(Area of Base x Height)
click to reveal
1
3
(Area of Base x Height)
Pyramids are named by the shape of their base..
3
The volume is a pyramid is 1/3 the volume of a prism
with the same base area( B) and height (h).
V = 1 Bh
3
1
V = 3 Bh
=5 m
si
de
le
n
h
gt
=
4
m
Answer
8 cm
Answer
24
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Slide 61 / 135
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.
28
Find the volume.
Answer
27
Slide 62 / 135
Answer
15.3 cm
10 in
7 cm
4 in
8 cm
8 in
Slide 63 / 135
Slide 64 / 135
Surface Area of Prisms
Surface Area
Return to
Table of
Contents
Return to
Table of
Contents
Slide 65 / 135
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Surface Area
Surface Area
The sum of the areas of all outside surfaces of a 3-D figure.
6 in
To find surface area, you must find the area of each surface of the
figure then add them together.
3 in
8 in
What type of figure is pictured?
6 in
How many surfaces are there?
How do you find the area of each
surface?
8 in
3 in
Bottom Top
8
8
x3
x3
24
24
18
6 in
48
+48
3 in
1808 inin2
Left Right
6
6
x3
x3
18
18
Front
8
x6
48
Back
SUM
8
x6
48 = 180 in 2
6 in
8 in
6 in
3 in
3 in
8 in
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Arrangement of Unit Cubes
Width
Height
Volume
for 6 arrangements
Length
Pull
Which arrangement of 24 cubes has the least
surface area?
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.
Surface
Sketch
Area
Pull
Which has the most?
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Arrangement of Unit Cubes
Width
Height
Volume
Pull
Surface
Sketch
Area
Which has the most?
Pull
Length
for 7 arrangements
Which arrangement of 64 cubes has the least
surface area?
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.
Slide 71 / 135
Which arrangement of 12 cubes has the least surface area?
A
1 x 1 x 27
A
2x2x3
B
3x3x3
B
4x3x1
C
9x3x1
C
2x6x1
D
1 x 1 x 12
Pull
30
Which arrangement of 27 cubes has the least surface area?
Pull
29
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A
1 x 1 x 25
B
1x5x5
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
3x8x2
E
4x2x6
F
4x3x4
Slide 75 / 135
Pull
32
Which arrangement of 25 cubes has the greatest surface
area?
Pull
31
Slide 74 / 135
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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.
Name the figure.
Find the figure's surface area.
Pull
5 in
7m
6 in
12 in
Top/Bottom
12
x 6
72
x 2
144
Front/Back
12
x5
60
x2
120
click to reveal
click to reveal
Left/Right
6
x5
30
x2
60
click to reveal
Surface Area
144
120
+ 60
324 in2
3m
5m
click to reveal
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33 How many faces does the figure have?
34
How many area problems must you complete when finding
the surface area?
Pull
6m
Pull
6m
2m
4m
2m
4m
Slide 79 / 135
35
Slide 80 / 135
What is the area of the top or bottom face?
What is the area of the left or right face?
36
Pull
6m
6m
Pull
2m
2m
4m
4m
Slide 81 / 135
37
Slide 82 / 135
What is the area of the front or back face?
What is the surface area of the figure?
38
Pull
Pull
6m
6m
2m
2m
4m
4m
Slide 83 / 135
Slide 84 / 135
Find the Surface Area.
1.
2.
3.
4.
5.
Draw and label ALL faces
Find the correct dimensions for each face
Calculate the AREA of EACH face
Find the SUM of ALL faces
Label Answer
5 yd
go on to see steps
5 yd
5 yd
5 yd
4 yd
6 yd
4 yd
9 yd
9 yd
6 yd
6 yd
Triangles
4
x6
CLICK TO REVEAL
24 / 2 = 12
x2
24
4 yd
Bottom Rectangle
9
CLICK TO REVEAL
x6
54
5 yd
9 yd
5 yd
5 yd
5 yd
4 yd
9 yd
6 yd
Left/Right Rectangles
(Same size since isosceles)
5
x 9 CLICK TO REVEAL
45
x2
90
Total
Surface
Area
CLICK TO REVEAL
24
54
+ 90
168 yd 2
Slide 85 / 135
Slide 86 / 135
Find the Surface Area.
1.
2.
3.
4.
5.
9 cm
9 cm
Draw and label ALL faces
Find the correct dimensions for each face
Calculate the AREA of EACH face
Find the SUM of ALL faces
Label Answer
9 cm
9 cm
6 cm
6 cm
11 cm
9 cm
Triangles
9 cm
11 cm
9 cm
Rectangles
Pull
TRY THIS ONE
9 cm
6 cm
11 cm
9 cm
Slide 87 / 135
39
Find the surface area of the shape below.
1.
2.
3.
4.
5.
Draw and label ALL faces
Find the correct dimensions for each face
Calculate the AREA of EACH face
Find the SUM of ALL faces
Label Answer
Slide 88 / 135
Find the Surface Area.
7 cm
4 cm
Pull
47 ft
15 cm
5 cm
50 ft
21 ft
3 cm
6 cm
42 ft
Slide 89 / 135
7 cm
Slide 90 / 135
3 cm
40
4 cm
8 cm
15 cm
6 cm
Pull
5 cm
Find the surface area of the shape below.
6 cm
Trapezoids
10
+6
16
x4
64 / 2 = 32
x 2
64 to reveal
click
Bottom Rectangle
6
x 15
90
Top Rectangle
10
x 15
150
click to reveal
click to reveal
Surface Area
64
90
150
+ 150
454 cm2
click to reveal
Side Rectangles
5
x 15
75
x 2
150
9 cm
click to reveal
10 cm
Slide 91 / 135
41
Slide 92 / 135
Find the surface area of the shape below.
2 cm
6c
m
Pull
c
10
6c
m
Surface Area of Pyramids
m
10 cm
Return to
Table of
Contents
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Slide 94 / 135
Surface Area of Pyramids
Find the Surface Area.
What is a pyramid?
Polyhedron with one base and triangular faces
click to reveal
that
meet at a vertex
17.5 cm
go on to see steps
How do you find Surface Area?
7 cm
Sum of the areas of all the surfaces of a 3-D
click to reveal
Figure
8 cm
Slide 95 / 135
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Find the Surface Area.
Bottom
Rectangle
8
x 7
56
Surface Area
56
140
+ 122.5
318.5 cm2
8 cm
Front/Back
Triangles
Find the surface area of a square pyramid with base edge of
4 inches and triangle height of 3 inches.
7 cm
7 cm
7 cm
8 cm
17.5 cm
17.5 cm
17.5 cm
3 in
8 cm
Left/Right
Triangles
Base
4
x4
16
click to reveal
4 in
4 Triangles
click to reveal
Surface Area
16
+ 24
40 in2
click to reveal
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Find the surface area.
42
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!).
6 in
A
Square Pyramid
B
Cube
Pull
Triangles
(all equal)
Base
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?
click to reveal
click to reveal
4 in
4 in
3.5 in
Surface Area
7
+ 36
43 in2
4 in
click to reveal
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44
43 Find the Surface Area of a triangular pyramid with
Find the Surface Area.
Pull
Pull
base edges of 8 in, base height of 4 in and a slant
height of 10 in.
11 m
10 in
12 m
8 in
8 in
6.9 in
6.7 m
9m
8 in
Slide 101 / 135
9m
Slide 102 / 135
How would you find the surface area of a cylinder?
Surface Area of Cylinders
Return to
Table of
Contents
Slide 103 / 135
Slide 104 / 135
Notice the length
of the rectangle is
actually the
circumference of
the circular base.
Radius
Radius
Curved Side = Circumference of Circular Base
H
E
I
G
H
T
H
E
I
G
H
T
Steps
1. 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.
Slide 105 / 135
Slide 106 / 135
Find the surface area of a cylinder whose height is 14 inches
and whose base has a diameter of 16 inches.
Radius
Radius
Curved Side = Circumference of Circular Base
H
E
I
G
H
T
H
E
I
G
H
T
16 in
14 in
Area of Circles = 2 ( r2 )
= 2 ( 82)
= 2 (64 )
= 128
= 401.92 in 2
Area of Curved Surface = Circumference Height
=
d Height
= (16)(14)
= 224
= 703.36 in2
Area of Circles = 2 ( r
2)
Area of Curved Surface = Circumference Height
=
dh
2
Surface Area = 401.92 + 703.36 = 1105.28 in
2 r + dh
-or2 r2 + 2 rh
2
Slide 108 / 135
45 Find the surface area of a cylinder whose height is
46
Pull
8 inches and whose base has a diameter of 6
inches.
Find the surface area of a cylinder whose height is 14
inches and whose base has a diameter of 16 inches.
Pull
Slide 107 / 135
How much material is needed to make a cylindrical orange
juice can that is 15 cm high and has a diameter of 10 cm?
48
Find the surface area of a cylinder with a height of 14
inches and a base radius of 8 inches.
Pull
47
Slide 110 / 135
Slide 111 / 135
49
Slide 112 / 135
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.
Pull
Yes
Surface Area of Spheres
No
Return to
Table of
Contents
Slide 113 / 135
A sphere is the set of all points that are the same
distance from the center point.
Slide 114 / 135
If the diameter of the Earth is 12,742 km, what is its
surface area?
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.
12,742 km
Surface Area of a Sphere
Radius
click to reveal
Pull
Slide 109 / 135
Slide 115 / 135
Slide 116 / 135
Try This:
50
Pull
Find the surface area of a tennis ball whose diameter is
2.7 inches.
Find the surface area of a softball with a diameter
3.8 inches.
2.7 in
click to reveal
Slide 117 / 135
52
How much leather is needed to make a basketball
with a radius of 4.7 inches?
How much rubber is needed to make 6
racquetballs with a diameter of 5.7 inches?
Pull
Pull
51
Slide 118 / 135
Slide 119 / 135
Slide 120 / 135
53
Find the volume.
22 mm
More Practice / Review
Pull
8 mm
15 mm
Return to
Table of
Contents
Slide 121 / 135
54
Slide 122 / 135
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.
55
Find the volume of a square pyramid with base edge of 4
inches and pyramid height of 3 inches.
HINT: Drawing a diagram will help!
Pull
Pull
Slide 123 / 135
56
Slide 124 / 135
Find the Volume
57
Find the Volume
Pull
Pull
21 ft
11 m
12 m
9m
6m
14 ft
9m
Slide 125 / 135
58
Slide 126 / 135
Find the Volume
59
Find the Volume
9 ft
Pull
Pull
8 in
6.9 in
8 ft
4 ft
7 ft
Slide 127 / 135
60
Slide 128 / 135
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 full of soil were needed to fill the planter?
61
Find the surface area of the cylinder.
Radius = 6 cm and Height = 7 cm
Pull
Pull
20 cm
14 cm
25 cm
Slide 129 / 135
62
Slide 130 / 135
Find the Surface Area.
63
11 cm
Find the Surface Area.
9 in
Pull
Pull
9 in
2 in
8 in
11 cm
7 in
12 cm
Slide 131 / 135
Slide 132 / 135
64 Find the Volume.
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?
9 in
9 in
Pull
2 in
8 in
7 in
Pull
Slide 133 / 135
Slide 134 / 135
67
66 Find the surface area of a square pyramid with a base
length of 4 inches and slant height of 5 inches.
Find the volume.
70 m
Pull
40 m
80 m
Slide 135 / 135
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?
Pull
Pull