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7th Grade Math
3D Geometry
2015-11-20
www.njctl.org
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Table of Contents
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
Surface Area
·
·
·
·
Prisms
Pyramids
Cylinders
Spheres
More Practice/ Review
Glossary & Standards
Slide 3 (Answer) / 159
Table of Contents
Click on the topic to
go to that section
Teacher Notes
3-Dimensional Solids
Cross Sections of 3-Dimensional Figures
Volume
· Prisms and Cylinders
Vocabulary Words are bolded
· Pyramids, Cones
Spheres
in &
the
presentation. The text
box the word is in is then
Surface Area
linked to the page at the end
· Prisms
of the presentation with the
· Pyramids
word defined on it.
· Cylinders
· Spheres
More Practice/ Review
Glossary & Standards
[This object is a pull tab]
Slide 4 / 159
3-Dimensional Solids
Return to
Table of
Contents
Slide 5 / 159
The following link will take you to a site with
interactive 3-D figures and nets.
Slide 6 / 159
Polyhedrons
Polyhedron
A 3-D figure whose faces are all polygons
Sort the figures into the appropriate side.
Polyhedron
Not Polyhedron
Slide 7 / 159
3-Dimensional Solids
Categories & Characteristics of 3-D Solids:
Prisms
1. Have 2 congruent, polygon bases which are parallel
to one another
click to reveal
2. Sides are rectangular (parallelograms)
3. Named by the shape of their base
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
Slide 8 / 159
3-Dimensional Solids
Categories & Characteristics of 3-D Solids:
Cylinders
1. Have 2 congruent, circular bases which
are parallel to one click
another
to reveal
2. Sides are curved
Cones
1. Have 1 circular base with a vertex opposite it
2. Sides are curvedclick to reveal
Slide 9 / 159
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
Point where 3 or more faces/edges meet
(pl. Vertices)
Slide 10 / 159
1
Name the figure.
A
Rectangular Prism
B
Triangular Pyramid
C
Hexagonal Prism
D
Rectangular Pyramid
E
Cylinder
F
Cone
Slide 10 (Answer) / 159
Name the figure.
A
Rectangular Prism
B
Triangular Pyramid
C
Hexagonal Prism
D
Rectangular Pyramid
E
Cylinder
F
Cone
Answer
1
D
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2
Name the figure.
A
B
C
D
E
F
Rectangular Pyramid
Triangular Prism
Octagonal Prism
Circular Pyramid
Cylinder
Cone
Slide 11 (Answer) / 159
Name the figure.
A
B
C
D
E
F
Rectangular Pyramid
Triangular Prism
Octagonal Prism
Circular Pyramid
Cylinder
Cone
Answer
2
E
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3
Name the figure.
A
B
C
D
E
F
Rectangular Pyramid
Triangular Pyramid
Triangular Prism
Hexagonal Pyramid
Cylinder
Cone
Slide 12 (Answer) / 159
Name the figure.
A
B
C
D
E
F
Rectangular Pyramid
Triangular Pyramid
Triangular Prism
Hexagonal Pyramid
Cylinder
Answer
3
Cone
B
[This object is a pull tab]
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4
Name the figure.
A
B
C
D
E
F
Rectangular Prism
Triangular Prism
Square Prism
Rectangular Pyramid
Cylinder
Cone
Slide 13 (Answer) / 159
Name the figure.
A
B
C
D
E
F
Rectangular Prism
Triangular Prism
Square Prism
Rectangular Pyramid
Cylinder
Cone
Answer
4
A
[This object is a pull tab]
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5
Name the figure.
A
B
C
D
E
F
Rectangular Prism
Triangular Pyramid
Circular Prism
Circular Pyramid
Cylinder
Cone
Slide 14 (Answer) / 159
Name the figure.
A
B
C
D
E
F
Rectangular Prism
Triangular Pyramid
Circular Prism
Circular Pyramid
Cylinder
Cone
Answer
5
F
[This object is a pull tab]
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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
3-Dimensional
Figures?
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
Name
Math Practice
Faces, Vertices and Edges
Slide 16 / 159
Euler's Formula
Euler's Formula:
E+2=F+V
The sum
of to
the
edges and 2 is
click
reveal
equal to the sum of the faces and vertices.
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6
How many faces does a pentagonal prism have?
Slide 17 (Answer) / 159
How many faces does a pentagonal prism have?
Answer
6
7
[This object is a pull tab]
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7
How many edges does a rectangular pyramid have?
Slide 18 (Answer) / 159
How many edges does a rectangular pyramid have?
Answer
7
8
[This object is a pull tab]
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8
How many vertices does a triangular prism have?
Slide 19 (Answer) / 159
How many vertices does a triangular prism have?
Answer
8
6
[This object is a pull tab]
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Cross Sections of
Three-Dimensional
Figures
Return to
Table of
Contents
Slide 21 / 159
Cross Sections
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, called a
cross section.
These 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.
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Cross Sections
A horizontal cross-section of a cone is a circle.
Can you describe a vertical cross-section of a cone?
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Cross Sections
A vertical cross-section of a cone is a triangle.
Slide 24 / 159
Cross Sections
A water tower is built in the shape of a cylinder.
How does the horizontal cross-section compare to the vertical
cross-section?
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Cross Sections
The horizontal cross-section is a circle.
The vertical cross-section is a rectangle
Slide 26 / 159
9
Which figure has the same horizontal and vertical
cross-sections?
A
C
B
D
Slide 26 (Answer) / 159
Which figure has the same horizontal and vertical
cross-sections?
C
A
Answer
9
B
C
D
[This object is a pull tab]
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10 Which figure does not have a triangle as one of its
cross-sections?
A
C
B
D
Slide 27 (Answer) / 159
10 Which figure does not have a triangle as one of its
cross-sections?
C
Answer
A
B
C
D
[This object is a pull tab]
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11 Which is the vertical cross-section of the figure
shown?
A
Triangle
B
Circle
C
Rectangle
D
Trapezoid
Slide 28 (Answer) / 159
Answer
11 Which is the vertical cross-section of the figure
shown?
A
C
Triangle
[This object is a pull tab]
B
Circle
C
Rectangle
D
Trapezoid
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12 Which is the horizontal cross-section of the figure
shown?
A
Triangle
B
Circle
C
Rectangle
D
Trapezoid
Slide 29 (Answer) / 159
A
Triangle
B
Circle
C
Rectangle
D
Trapezoid
Answer
12 Which is the horizontal cross-section of the figure
shown?
C
[This object is a pull tab]
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13 Which is the vertical cross-section of the figure
shown?
A
Triangle
B
Circle
C
Square
D
Trapezoid
Slide 30 (Answer) / 159
Answer
13 Which is the vertical cross-section of the figure
shown?
A
Triangle
B
Circle
C
Square
D
Trapezoid
A
[This object is a pull tab]
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14 Misha has a cube and a right-square pyramid that are
made of clay. She placed both clay figures on a flat
surface.
Select each choice that identifies the two-dimensional
plane sections that could result from a vertical or
horizontal slice through each clay figure.
A Cube cross section is a Triangle
B Cube cross section is a Square
C Cube cross section is a Rectangle (not a square)
D Right-Square Pyramid cross section is a Triangle
E Right-Square Pyramid cross section is a Square
F Right-Square Pyramid cross section is a
Rectangle (not a square)
From PARCC EOY sample test calculator #11
Slide 31 (Answer) / 159
14 Misha has a cube and a right-square pyramid that are
made of clay. She placed both clay figures on a flat
surface.
Select each choice that identifies the two-dimensional
plane sections that could result from a vertical or
horizontal slice through each
C clay figure.
B
A Cube cross section is a Triangle
Answer
B Cube cross section is a Square
E
D
C Cube cross section is a Rectangle (not a square)
D Right-Square Pyramid cross section is a Triangle
Note: other angles for vertical cross
sections Pyramid
are possible,
you will
E Right-Square
crossbut
section
is a Square
object
still get the[This
same
2-D shapes
is a pullcross
tab] section is a
F Right-Square Pyramid
Rectangle (not a square)
From PARCC EOY sample test calculator #11
Slide 32 / 159
Volume
Return to
Table of
Contents
Slide 33 / 159
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)
click to reveal
Label
click
tocubic
revealunits
3
Units
or
Slide 33 (Answer) / 159
Volume
Volume
Label
Math Practice
- The amount of space occupied by a 3-D Figure
MP6:
Attend
to precision.
- The number of cubic
units
needed
to FILL a 3-D Figure (layering)
click to reveal
Continuously emphasize the units used
to label the answers.
Ask: What labels should we use?
click
tocubic
revealunits
3
Units
or
[This object is a pull tab]
Slide 34 / 159
Volume Activity
Click the link below for the activity
Lab #1: Volume Activity
Slide 34 (Answer) / 159
Volume Activity
Math Practice
Click the link
below
for the activity
This
lab addresses
MP7: Look for and make use of
structure
LabLook
#1: Volume
Activityregularity in
MP8:
for and express
repeated reasoning.
Ask: Do you see a pattern? Can you
explain it? (MP7 & MP8)
Can you predict the next one? (MP7 &
MP8)
[This object is a pull tab]
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Volume of Prisms & Cylinders
Return to
Table of
Contents
Slide 36 / 159
Volume
Volume of Prisms & Cylinders:
click
Area
of Base x Height
_____________________________________
Area Formulas:
Rectangle = lw
bh
clickor ______
Triangle = bh or
(bh)
click
2 ______
Circle =
r2
click
______
Slide 37 / 159
Volume
Find the Volume.
8m
2m
5m
Slide 37 (Answer) / 159
Volume
Answer
Find the Volume.
VOLUME:
2
x5
10 (Area of Base)
x 8 (Height)
8
m m3
80
VOLUME:
V=B h
V=l w h
V=5 2 8
V = 10 8
V = 80 m3
[This object is a pull tab]
2m
5m
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Volume
Find the Volume. Use 3.14 as your value of π.
9 yd
10 yd
Slide 38 (Answer) / 159
Volume
Answer
Find the Volume. Use 3.14 as your value of π.
VOLUME:
VOLUME:
9
V=B h
x9
V = r2 h
81
V = 3.14 92 10
x
3.14
9 yd
254.34 (Area of Base) V = 3.14 81 10
V = 254.34 10
x 10 (Height)
2543.4 yd3
V = 2543.4 yd3
10 yd
[This object is a pull tab]
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15 Find the volume.
4 in
7 1 in
5
1 1 in
2
Slide 39 (Answer) / 159
Answer
15 Find the volume.
7 1 in
5
4 in
1 1 in
2
VOLUME:
7.2
x 1.5
10.8 (Area of Base)
x 4 (Height)
43.2 in3
VOLUME:
V=B h
V = 7.2(1.5)(4)
V = 43.2 in3
[This object is a pull tab]
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16 Find the volume of a rectangular prism with length 2
cm, width 3.3 cm and height 5.1 cm.
Slide 40 (Answer) / 159
Answer
16 Find the volume of a rectangular prism with length 2
cm, width 3.3 cm and height 5.1 cm.
VOLUME:
V=B h
V =2(3.3)(5.1)
V = (6.6)(5.1)
V = 33.66 cm3
[This object is a pull tab]
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17 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
Slide 41 (Answer) / 159
A
1 x 2 x 18
B
6x3x3
C
2x3x3
D
3x3x3
Answer
17 Which is a possible length, width and height for a
rectangular prism whose volume = 18 cm 3?
C
[This object is a pull tab]
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18 Find the volume.
Slide 42 (Answer) / 159
Answer
18 Find the volume.
[This object is a pull tab]
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19 Find the volume. Use 3.14 as your value of π.
10 m
6m
Slide 43 (Answer) / 159
19 Find the volume. Use 3.14 as your value of π.
6m
Answer
10 m
[This object is a pull tab]
Slide 44 / 159
Teachers:
Use this Mathematical Practice Pull Tab for the next 3
SMART Response slides.
Slide 44 (Answer) / 159
The next 4 slides address MP4 & MP5
Math Practice
Ask: What connections do you see
Teachers: between the volume of a prism/cylinder
and this problem?
Use this Mathematical
Practice(MP4)
Pull Tab for the next 3
SMART Response
slides.
Could you
use a drawing to show your
thinking? (MP5)
Why do the results make sense? (MP4)
[This object is a pull tab]
Slide 45 / 159
20 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!
Slide 45 (Answer) / 159
20 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
HINT: You may want to draw a picture!
[This object is a pull tab]
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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?
Use 3.14 as your value of π.
Slide 46 (Answer) / 159
Answer
21 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.
[This object is a pull tab]
Slide 47 / 159
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 as your value of π.
Slide 47 (Answer) / 159
Answer
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 as your value of π.
[This object is a pull tab]
Slide 48 / 159
Teachers:
Use this Mathematical Practice Pull Tab for the next
SMART Response slide.
Slide 48 (Answer) / 159
Teachers:
Math Practice
Use this Mathematical Practice Pull Tab for the next
SMART Response slide.
The next slide addresses MP3
Ask: What math language will help you
prove your answer?
Why is it true?
[This object is a pull tab]
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23 Which circular glass holds more water?
Use 3.14 as your value of π. Before
revealing your answer, make sure that you
can prove that your answer is correct.
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
Slide 49 (Answer) / 159
Answer
23 Which circular glass holds more water?
Use 3.14 as your value of π. Before
revealing your answer, make
sure
Glass
A that you Glass B
can prove that your answer is correct.
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
[This object is a pull tab]
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Volume of Pyramids, Cones &
Spheres
Return to
Table of
Contents
Slide 51 / 159
Slide 51 (Answer) / 159
Slide 52 / 159
Demonstration Comparing Volume of
Cones & Spheres with Volume of
Cylinders
click to go to web site
Slide 53 / 159
Volume of a Cone
The Volume of 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 (Area
click to
ofreveal
Base x Height)
3
3
Slide 54 / 159
Volume of a Sphere
The Volume of a
Sphere is 2/3 the
volume of a
cylinder with the
same base area
(B) and height (h).
V = 2/3 (Volume of Cylinder)
V= 2/3 (
or
V = 4/3
r2click
h )to reveal
r3
Slide 55 / 159
Volume
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)
Slide 55 (Answer) / 159
Volume
Answer
3.14
9
How much ice cream canxa Friendly’s
Waffle cone hold if
it has a diameter of 6 in and
its height
10Base)
in?
28.26
(Areaisof
x 10 (Height)
(Just Ice Cream within Cone. Not on Top)
282.6
3
(cone)
3
= 94.2[This
inobject
is a pull tab]
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24 Find the volume.
9 in
4 in
Slide 56 (Answer) / 159
9 in
Answer
24 Find the volume.
[This object is a pull tab]
4 in
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25 Find the volume.
5 cm
8 cm
Slide 57 (Answer) / 159
Answer
25 Find the volume.
5 cm
8 cm
[This object is a pull tab]
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Slide 59 / 159
26 What is the volume of a sphere with a radius of 8 ft?
Slide 59 (Answer) / 159
Answer
26 What is the volume of a sphere with a radius of 8 ft?
[This object is a pull tab]
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27 What is the volume of a sphere with a diameter of
4.25 in?
Slide 60 (Answer) / 159
Answer
27 What is the volume of a sphere with a diameter of
4.25 in?
[This object is a pull tab]
Slide 61 / 159
Volume of a Pyramid
The Volume of a
Pyramid is 1/3 the
volume of a prism
with the same base
area (B) and height
(h).
(Area of Base x Height) 3
click to reveal
1
3 (Area of Base x Height)
Note: Pyramids are named by the shape of their base.
Slide 62 / 159
Volume
Example: Find the volume of the pyramid shown below.
1
V = 3 Bh
1
V = 3 Bh
=5 m
e
d
i
s
len
h
gt
=
4
m
Slide 63 / 159
28 Find the Volume of a triangular pyramid with base
edges of 8 in, base height of 6.9 in and a pyramid
height of 10 in.
10 in
6.9 in
8 in
Slide 63 (Answer) / 159
28 Find the Volume of a triangular pyramid with base
edges of 8 in, base height of 6.9 in and a pyramid
height of 10 in.
Answer
V = 1 Bh
3
10 in
6.9 in
8 in
V = 1 1 (8)(6.9) (10)
3 2
V = 1 [27.6](10)
3
V = 1 (276)
3
V = [This
92 object
in3 is a pull tab]
Slide 64 / 159
29 Find the volume.
15.3 cm
7 cm
8 cm
Slide 64 (Answer) / 159
Answer
29 Find the volume.
15.3 cm
7 cm
8 cm
[This object is a pull tab]
Slide 65 / 159
Surface Area
Return to
Table of
Contents
Slide 66 / 159
Surface Area of Prisms
Return to
Table of
Contents
Slide 67 / 159
Surface Area
Surface Area is the 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.
What type of figure is pictured?
6 in
How many surfaces are there?
How do you find the area of each
surface?
3 in
8 in
Slide 68 / 159
Surface Area
6 in
3 in
8 in
Bottom Top
8
8
x3
x3
24
24
18
6 in
48
3 in
+488 in
180 in2
Left Right
6
6
x3
x3
18
18
Front
8
x6
48
Back SUM
8
x6
48 = 180 in 2
6 in
6 in
3 in
8 in
3 in
8 in
Slide 69 / 159
Slide 70 / 159
Slide 71 / 159
Arrangement of Unit Cubes
Surface Area Activity
Click the link below for the activity
Lab #2: Surface Area Activity
Slide 72 / 159
Teachers:
Use this Mathematical Practice Pull Tab for the next 4
SMART Response slides.
Slide 72 (Answer) / 159
The next 4 slides address MP2
Math Practice
Teachers:
Ask: HowPractice
can youPull
represent
Use this Mathematical
Tab forthe
the next 4
problem
w/ symbols and numbers?
SMART Response
slides.
What do you think the answer/result will
be?
[This object is a pull tab]
Slide 73 / 159
30 Which arrangement of 27 cubes has the least surface
area?
A
1 x 1 x 27
B
3x3x3
C
9x3x1
Slide 73 (Answer) / 159
A
1 x 1 x 27
B
3x3x3
C
9x3x1
Answer
30 Which arrangement of 27 cubes has the least surface
area?
B
[This object is a pull tab]
Slide 74 / 159
31 Which arrangement of 12 cubes has the least surface
area?
A
2x2x3
B
4x3x1
C
2x6x1
D
1 x 1 x 12
Slide 74 (Answer) / 159
A
2x2x3
B
4x3x1
C
2x6x1
D
1 x 1 x 12
Answer
31 Which arrangement of 12 cubes has the least surface
area?
A
[This object is a pull tab]
Slide 75 / 159
32 Which arrangement of 25 cubes has the greatest
surface area?
A
1 x 1 x 25
B
1x5x5
Slide 75 (Answer) / 159
A
1 x 1 x 25
B
1x5x5
Answer
32 Which arrangement of 25 cubes has the greatest
surface area?
A
[This object is a pull tab]
Slide 76 / 159
33 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 76 (Answer) / 159
A
4 x 12 x 1
B
2 x 2 x 12
C
1 x 1 x 48
D
3x8x2
E
4x2x6
F
4x3x4
Answer
33 Which arrangement of 48 cubes has the least surface
area?
F
[This object is a pull tab]
Slide 77 / 159
Slide 78 / 159
Slide 78 (Answer) / 159
Slide 79 / 159
34 How many faces does the figure have?
6m
2m
4m
Slide 79 (Answer) / 159
Answer
34 How many faces does the figure have?
6m
2m
6
[This object is a pull tab]
4m
Slide 80 / 159
35 How many area problems must you complete when
finding the surface area?
6m
2m
4m
Slide 80 (Answer) / 159
Answer
35 How many area problems must you complete when
finding the surface area?
6m
2m
4m
6
3 (if you double)
[This object is a pull tab]
Slide 81 / 159
36 What is the area of the top or bottom face?
6m
2m
4m
Slide 81 (Answer) / 159
Answer
36 What is the area of the top or bottom face?
6m
(2)(4)
8 m2
[This object is a pull tab]
2m
4m
Slide 82 / 159
37 What is the area of the left or right face?
6m
2m
4m
Slide 82 (Answer) / 159
Answer
37 What is the area of the left or right face?
6m
2m
(2)(6)
12 m2
[This object is a pull tab]
4m
Slide 83 / 159
38 What is the area of the front or back face?
6m
2m
4m
Slide 83 (Answer) / 159
Answer
38 What is the area of the front or back face?
6m
(6)(4)
24 m2
[This object is a pull tab]
2m
4m
Slide 84 / 159
Slide 84 (Answer) / 159
Slide 85 / 159
Find the Surface Area
1.
2.
3.
4.
5.
Draw and label ALL faces; use the net, if it's helpful
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
4 yd
9 yd
6 yd
Slide 86 / 159
Surface Area
5 yd
5 yd
5 yd
5 yd
4 yd
4 yd
9 yd
9 yd
6 yd
6 yd
Triangles
4
x6
CLICK TO REVEAL
24 / 2 = 12
x2
24
Bottom Rectangle
9
CLICK TO REVEAL
x6
54
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 87 / 159
Find the Surface Area Using the Net
5 yd
5 yd
4 yd
9 yd
6 yd
Triangles
4
x6
CLICK TO REVEAL
24 / 2 = 12
x2
24
Middle Rectangle
9CLICK TO REVEAL
x6
54
Left/Right
CLICK Rectangles
TO REVEAL
(Same size since isosceles)
5 x 9 = 45 x 2 = 90
Total
Surface
Area
CLICK TO REVEAL
24
54
+ 90
168 yd 2
Slide 88 / 159
Find the Surface Area
1.
2.
3.
4.
5.
Draw and label ALL faces; use the net if it's helpful
Find the correct dimensions for each face
Calculate the AREA of EACH face
Find the SUM of ALL faces
Label Answer
9 cm
TRY THIS ONE
9 cm
7.8 cm
9 cm
11 cm
Slide 89 / 159
Surface Area
9 cm
9 cm
9 cm
9 cm
7.8 cm
9 cm
Triangles
A = 7.8 x 9
2
A = 35.1 cm2
x 2
70.2 cm2
11 cm
7.8 cm
9 cm
Rectangles
A = 9(11) = 99 cm2
A = 99 x 3 = 297 cm2
11 cm
Slide 89 (Answer) / 159
Surface Area
9 cm
Answer
9 cm
9 cm
9 cm
7.8 cm
9 cm
Triangles
A = 7.8 x 9
2
A = 35.1 cm2
x 2
70.2 cm2
11 cm
367.2 cm2
7.8 cm
9 cm
11 cm
Rectangles
A = 9(11) = 99 cm2
[This object is a pull tab]
A = 99 x 3 = 297 cm2
Slide 90 / 159
Surface Area
9 cm
9 cm
7.8 cm
11 cm
9 cm
Triangles
A = 7.8 x 9
2
A = 35.1 cm2
x 2
70.2 cm2
click to reveal
Rectangles
A = 9(11) = 99 cm2
Aclick
= 99
x 3 = 297 cm2
to reveal
Slide 90 (Answer) / 159
Surface Area
9 cm
7.8 cm
11 cm
Answer
9 cm
367.2 cm2
9 cm
Triangles
A = 7.8 x 9
2
A = 35.1 cm2
x 2
70.2 cm2
click to reveal
[This object is a pull tab]
Rectangles
A = 9(11) = 99 cm2
Aclick
= 99
x 3 = 297 cm2
to reveal
Slide 91 / 159
40 Find the surface area of the shape below.
1. Draw and label ALL faces; use the net if it's
helpful
2. Find the correct dimensions for each face
3. Calculate the AREA of EACH face
4. Find the SUM of ALL faces
5. Label Answer
47 ft
50 ft
21 ft
42 ft
Slide 91 (Answer) / 159
40 Find the surface area of the shape below.
Answer
1. Draw and label ALL faces; use the net if it's
helpful
2. Find the correct dimensions for each face
3. Calculate the AREA of EACH face 6,382 ft2
4. Find the SUM of ALL faces
5. Label Answer
47 ft
[This object is a pull tab]
50 ft
21 ft
42 ft
Slide 92 / 159
Find the Surface Area.
9 cm
4 cm
15 cm
5 cm
6 cm
3 cm
Slide 93 / 159
9 cm
3 cm
4 cm
15 cm
5 cm
6 cm
Trapezoids
12
+6
18
x4
72 / 2 = 36
x2
72
click to reveal
Bottom Rectangle
6
x 15
90
Top Rectangle
12
x 15
180
click to reveal
click to reveal
Side Rectangles
5
x 15
75
x 2
150
click to reveal
Slide 93 (Answer) / 159
9 cm
3 cm
4 cm
5 cm
6 cm
Trapezoids
12
+6
18
x4
72 / 2 = 36
x2
72
click to reveal
Answer
15 cm
Bottom Rectangle
6
x 15
90
Surface Area
72
90
180
Top Rectangle
Side Rectangles
12 + 150
5
x 15 492 cm2 x 15
180
75
x 2
[This object is a pull tab]
150
click to reveal
click to reveal
click to reveal
Slide 94 / 159
41 Find the surface area of the shape below.
6 cm
8 cm
9 cm
10 cm
Slide 94 (Answer) / 159
41 Find the surface area of the shape below.
6
Answer
8 cm
Bases Sides
cm2 Triangles3 Rectangles
9 cm
[This object is a pull tab]
10 cm
Slide 95 / 159
42 Find the surface area of the shape below.
2 cm
cm
0
1
6c
m
10 cm
m
6c
Slide 95 (Answer) / 159
of the shape below.
42 Find the surface area
Bases Sides
Square & Rectangle
2 cm
6 Rectangles
6c
Answer
cm
0
1
m
m
6c
10 cm
[This object is a pull tab]
Slide 96 / 159
Surface Area of Pyramids
Return to
Table of
Contents
Slide 97 / 159
Surface Area of Pyramids
What is a pyramid?
Polyhedron with one base and triangular faces
that meet at a vertex click to reveal
How do you find Surface Area?
Sum of the areas of all the surfaces of a 3-D
Figure
click to reveal
Slide 98 / 159
Find the Surface Area.
17.5 cm
go on to see steps
17.4 cm
8 cm
7 cm
Slide 99 / 159
Find the Surface Area.
Bottom
Rectangle
8
x 7
56 cm2
Front/Back
Triangles
A = 1 bh(2)
2
A = 1 (8)(17.4)(2)
2
A = 139.2 cm2
Left/Right
Triangles
A = 1 bh(2)
2
A = 1 (7)(17.5)(2)
2
A = 122.5 cm2
Slide 99 (Answer) / 159
Bottom
Rectangle
8
x 7
56 cm2
Answer
Find the Surface Area.
Front/Back
Triangles
Surface Area
56
139.2 Left/Right
+ 122.5Triangles
317.7Acm
= 21 bh(2)
A = 1 bh(2)
2
A = 1 (8)(17.4)(2)
2
A = 139.2 cm2
2
A = 1 (7)(17.5)(2)
[This object is a pull tab]
2
A = 122.5 cm2
Slide 100 / 159
Surface Area
Find the surface area of a square pyramid with base edge of
4 inches and triangle height of 3 inches.
3 in
Base
4
x4
16
click to reveal
4 in
4 Triangles
click to reveal
Surface Area
16
+ 24
40 in2
click to reveal
Slide 101 / 159
Surface Area
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
Remaining
equivalent!).
Triangles
Base
(all equal)
6 in
4 in
4 in
3.5 in
click to reveal
click to reveal
4 in
Surface Area
7
+ 36
43 in2
click to reveal
Slide 101 (Answer) / 159
Surface Area
Hint
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
Remaining
equivalent!).
Triangles
Base
(all equal)
6 in
4 in
4 in
3.5 in
[This object is a pull tab] click to reveal
click to reveal
4 in
Surface Area
7
+ 36
43 in2
click to reveal
Slide 102 / 159
43 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
B
Square Pyramid
Cube
Slide 102 (Answer) / 159
A
B
Answer
43 Which has a greater Surface Area, a square pyramid
with a base edge of 8 inSquare
andPyramid
a height of 4 in or Cube
a cube
with an edge of 5 in?
Square Pyramid
Cube
SA = 128 in2
[This object is a pull tab]
B
Slide 103 / 159
44 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.
10 in
8 in
6.9 in
8 in
8 in
Slide 103 (Answer) / 159
Answer
44 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.
10 in
8 in
6.9 in
8 in
[This object is a pull tab]
8 in
Slide 104 / 159
45 Find the Surface Area.
11 m
12 m
6.7 m
9m
9m
Slide 104 (Answer) / 159
12 m
6.7 m
9m
Answer
45 Find the Surface Area.
11 m
9m
[This object is a pull tab]
Slide 105 / 159
Surface Area of Cylinders
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Table of
Contents
Slide 106 / 159
Surface Area
How would you find the surface area of a cylinder?
Slide 107 / 159
Surface Area
Notice the length
of the rectangle is
actually the
circumference of
the circular base.
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 108 / 159
Surface Area
Radius
H
E
I
G
H
T
Middle step to get to the
net
Original cylinder
Radius
Curved Side = Circumference of Circular Base
H
E
I
G
H
T
Net of a cylinder
Slide 109 / 159
Surface Area
Radius
Radius
H
E
I
G
H
T
Curved Side = Circumference of Circular Base
H
E
I
G
H
T
Area of Circles = 2 (πr2)
Area of Curved Surface = Circumference Height
=πd h
2πr2 + πdh
-or-
2πr2 + 2πrh
Slide 110 / 159
Surface Area
Find the surface area of a cylinder whose height is 14 inches
and whose base has a diameter of 16 inches. Use 3.14 as your value
of π.
16 in
14 in
Area of Circles = 2 (πr2 )
= 2 (π82)
= 2 (64π)
= 128π
= 401.92 in2
Area of Curved Surface = Circumference Height
= π d Height
= π(16)(14)
= 224π
= 703.36 in2
2
Surface Area = 401.92 + 703.36 = 1,105.28 in
Slide 111 / 159
46 Find the surface area of a cylinder whose height is 8
inches and whose base has a diameter of 6
inches. Use 3.14 as your value of π.
Slide 111 (Answer) / 159
Answer
46 Find the surface area of a cylinder whose height is 8
inches and whose base has a diameter of 6
inches. Use 3.14 as your value of π.
[This object is a pull tab]
Slide 112 / 159
47 Find the surface area of a cylinder whose height is 14
inches and whose base has a diameter of 20 inches.
Use 3.14 as your value of π.
Slide 112 (Answer) / 159
Answer
47 Find the surface area of a cylinder whose height is 14
inches and whose base has a diameter of 20 inches.
Use 3.14 as your value of π.
Circles
A = πr2(2)
A = π(10)2(2)
A = 628 in2
Side
A = πdh
A = π(20)(14)
A = 879.2 in2
SA = 628 + 879.2 = 1,507.2 in2
[This object is a pull tab]
Slide 113 / 159
48 How much material is needed to make a cylindrical
orange juice can that is 15 cm high and has a diameter
of 10 cm? Use 3.14 as your value of π.
Slide 113 (Answer) / 159
Answer
48 How much material is needed to make a cylindrical
orange juice can that is 15 cm high and has a diameter
of 10 cm? Use 3.14 as your value of π.
[This object is a pull tab]
Slide 114 / 159
49 Find the surface area of a cylinder with a height of 14
inches and a base radius of 8 inches. Use 3.14 as
your value of π.
Slide 114 (Answer) / 159
Answer
49 Find the surface area of a cylinder with a height of 14
inches and a base radius of 8 inches. Use 3.14 as
your value of π.
[This object is a pull tab]
Slide 115 / 159
50 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. Note: Use 3.14 as your
value of π.
Yes
No
Slide 115 (Answer) / 159
Yes
No
Answer
50 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. Note: Use 3.14 as your
value of π.
NO
[This object is a pull tab]
Slide 116 / 159
Surface Area of Spheres
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Table of
Contents
Slide 117 / 159
Surface Area
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 π.
Surface Area of a Sphere
Radius
click to reveal
Slide 118 / 159
Surface Area
If the diameter of the Earth is 12,742 km, what is its
surface area? Use 3.14 as your value of π. Round your
answer to the nearest whole number.
12,742 km
Slide 119 / 159
Surface Area
Try This:
Find the surface area of a tennis ball whose diameter is
2.7 inches. Use 3.14 as your value of π.
2.7 in
click to reveal
Slide 120 / 159
51 Find the surface area of a softball with a diameter
3.8 inches. Use 3.14 as your value of π.
Slide 120 (Answer) / 159
Answer
51 Find the surface area of a softball with a diameter
3.8 inches. Use 3.14 as your value of π.
[This object is a pull tab]
Slide 121 / 159
52 How much leather is needed to make a basketball
with a radius of 4.7 inches? Use 3.14 as your value of π.
Slide 121 (Answer) / 159
Answer
52 How much leather is needed to make a basketball
with a radius of 4.7 inches? Use 3.14 as your value of π.
[This object is a pull tab]
Slide 122 / 159
53 How much rubber is needed to make 6
racquet balls with a diameter of 5.7 inches?
Use 3.14 as your value of π.
Slide 122 (Answer) / 159
Answer
53 How much rubber is needed to make 6
racquet balls with a diameter of 5.7 inches?
Use 3.14 as your value of π.
[This object is a pull tab]
Slide 123 / 159
More Practice / Review
Return to
Table of
Contents
Slide 124 / 159
54 Find the volume.
22 mm
8 mm
15 mm
Slide 124 (Answer) / 159
22 mm
Answer
54 Find the volume.
880 mm3
8 mm
15 mm
[This object is a pull tab]
Slide 125 / 159
55 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!
Slide 125 (Answer) / 159
55 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.
Answer
HINT: Drawing a diagram will help!
2.808 m3
[This object is a pull tab]
Slide 126 / 159
56 Find the volume of a square pyramid with base edge
of 4 inches and pyramid height of 3 inches.
Slide 126 (Answer) / 159
Answer
56 Find the volume of a square pyramid with base edge
of 4 inches and pyramid height of 3 inches.
16 in3
[This object is a pull tab]
Slide 127 / 159
57 Find the Volume.
11 m
12 m
9m
6m
9m
Slide 127 (Answer) / 159
Answer
57 Find the Volume.
148.5 m3
11 m
[This object is a pull tab]
12 m
9m
6m
9m
Slide 128 / 159
58 Find the Volume. Use 3.14 as your value of π.
21 ft
14 ft
Slide 128 (Answer) / 159
21 ft
Answer
58 Find the Volume. Use 3.14 as your value of π.
1077.02 ft3
[This object is a pull tab]
14 ft
Slide 129 / 159
59 Find the Volume. Use 3.14 as your value of π.
8 in
6.9 in
Slide 129 (Answer) / 159
Answer
59 Find the Volume. Use 3.14 as your value of π.
6.9 in
8 in
115.552 in3
[This object is a pull tab]
Slide 130 / 159
60 Find the volume.
8.06 ft
8 ft
4 ft
7 ft
Slide 130 (Answer) / 159
60 Find the volume.
Answer
8.06 ft
112 ft3
8 ft
4 ft
7 ft
[This object is a pull tab]
Slide 131 / 159
61 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? Use 3.14 as your value of π.
20 cm
14 cm
25 cm
Slide 131 (Answer) / 159
61 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? Use 3.14 as your value of π.
Answer
20 cm
10.663 or
about 11
cones full
14 cm
[This object is a pull tab]
25 cm
Slide 132 / 159
62 Find the surface area of the cylinder. Use 3.14
as your value of π.
Radius = 6 cm and Height = 7 cm
Slide 132 (Answer) / 159
62 Find the surface area of the cylinder. Use 3.14
as your value of π.
Answer
Radius = 6 cm and Height = 7 cm
489.84 cm2
[This object is a pull tab]
Slide 133 / 159
63 Find the Surface Area.
11 cm
11 cm
12 cm
Slide 133 (Answer) / 159
63 Find the Surface Area.
Answer
11 cm
770 cm2
11 cm
[This object is a pull tab]
12 cm
Slide 134 / 159
64 Find the Surface Area.
9 in
9 in
8.3 in
8 in
7 in
Slide 134 (Answer) / 159
64 Find the Surface Area.
9 in
Answer
9 in
8.3 in
258.1 in2
8 in
7 in
[This object is a pull tab]
Slide 135 / 159
65 Find the volume.
9 in
9 in
8.3 in
8 in
7 in
Slide 135 (Answer) / 159
65 Find the volume.
9 in
8.3 in
8 in
Answer
9 in
232.4 in3
7 in
[This object is a pull tab]
Slide 136 / 159
66 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?
Slide 136 (Answer) / 159
Answer
66 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?
846 in2
[This object is a pull tab]
Slide 137 / 159
67 Find the surface area of a square pyramid with a
base length of 4 inches and slant height of 5
inches.
Slide 137 (Answer) / 159
Answer
67 Find the surface area of a square pyramid with a
base length of 4 inches and slant height of 5
inches.
56 in2
[This object is a pull tab]
Slide 138 / 159
68 Find the volume.
70 m
40 m
80 m
Slide 138 (Answer) / 159
68 Find the volume.
Answer
70 m
224,000 m3
40 m
80 m
[This object is a pull tab]
Slide 139 / 159
69 A teacher made 2 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?
Slide 139 (Answer) / 159
Answer
69 A teacher made 2 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?
1,200 in2
[This object is a pull tab]
Slide 140 / 159
Glossary & Standards
Return to
Table of
Contents
Teacher Notes
Slide 140 (Answer) / 159
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.
Glossary & Standards
[This object is a pull tab]
Return to
Table of
Contents
Slide 141 / 159
Cone
A 3-D solid that has 1 circular base
with a vertex opposite it.
The sides are curved.
Back to
Instruction
Slide 142 / 159
Cross Section
The shape formed when cutting
straight through an object.
Triangle
Square
Trapezoid
Back to
Instruction
Slide 143 / 159
Cylinder
A solid that has 2 congruent, circular bases
which are parallel to one another.
The side joining the 2 circular bases is a
curved rectangle.
shapes that form
a cylinder
top
folding down the
2 circles & rolling
the rectangle
Cylinder
bottom
Back to
Instruction
Slide 144 / 159
Face
Flat surface of a Polyhedron.
1 face
1 face
1 face
Back to
Instruction
Slide 145 / 159
Edge
Line segment formed where 2 faces meet.
1 edge
1 edge
1 edge
Back to
Instruction
Slide 146 / 159
Euler's Formula
The sum of the edges and 2 is equal
to the sum of the faces and vertices.
E+2=F+V
pyramid:
E+2=F+V
vertices = 4
E+2=4+4
E+2=8
E=6
faces = 4
Back to
Instruction
Slide 147 / 159
Net
A 2-D pattern of a 3-D solid that can be
folded to form the figure. An unfolded
geometric solid.
Solid
Solid
Net
Net
Solid
Net
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Instruction
Slide 148 / 159
Polyhedron
A 3-D figure whose faces are all polygons.
A Polyhedron has NO curved surfaces.
Plural: Polyhedra
Yes, Polyhedron
Yes, Polyhedron
No, not a
polyhedron
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Instruction
Slide 149 / 159
Prism
A polyhedron that has 2 congruent, polygon
bases which are parallel to one another.
Remaining sides are rectangular
(parallelograms). Named by the shape of the
base.
Triangular
Prism
Hexagonal
Prism
Octagonal
Prism
Back to
Instruction
Slide 150 / 159
Pyramid
A polyhedron that has 1 polygon base with a
vertex opposite it. Remaining sides are
triangular. Named by the shape of their base
Pentagonal
Pyramid
Rectangular
Pyramid
Triangular
Pyramid
Back to
Instruction
Slide 151 / 159
Surface Area
The sum of the areas of all outside
surfaces of a 3-D figure.
5
3
4
6
5
3 3 6 u2
5
4
6
24 u2 30 u2 6
4
2
5
3 6 u5
18 u2
1. Find the area
of each surface
of the figure
2. Add all of the
areas together
18
24
30
6
+6
SA = 84 units2
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Instruction
Slide 152 / 159
Vertex
Point where 3 or more faces/edges meet
Plural: Vertices
1 vertex
1 vertex
1 vertex
Back to
Instruction
Slide 153 / 159
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:
V=?
cylinder filled
halfway
with water
V=?
Prism filled
with water
Units3
or
cubic units
Back to
Instruction
Slide 154 / 159
Volume of a Cone
A cone is 1/3 the volume of a cylinder with the
same base area (B = πr2) and height (h).
V = (πr2h) ÷ 3
h
or
V=
1
3
πr2h
r
V = 1 πr2h
3
6
4
V = /3π(4)2(6)
V = 32π units3
V = 100.48 u3
1
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Instruction
Slide 155 / 159
Volume of a Cylinder
Found by multiplying the Area of the
base (B) and the height (h).
Since your
base is always a
circle, your volume
formula for a
cylinder is
V = Bh
V = πr2h
4
r
h
V = πr2h
10
V = π(4)2(10)
V = 160π units3
V = 502.4 u3
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Instruction
Slide 156 / 159
Volume of a Prism
Found by multiplying the Area of the
base (B) and the height (h).
V = Bh
The shape
of your base
matches the
name of the
prism
Rectangular
Prism
Triangular
Prism
V = Bh
V = (1/2b∆h∆)hprism
V = Bh
V = (lw)h
Back to
Instruction
Slide 157 / 159
Volume of a Pyramid
A pyramid is 1/3 the volume of a prism with
the same base area (B) and height (h).
V = (Bh) ÷ 3
The shape
of your base
matches the
name of the
pyramid
or
V = 1 (Bh)
3
Rectangular
Pyramid
V = 1/3Bh
V = 1/3(lw)h
Triangular
Pyramid
V = 1/3Bh
V = (1/2b∆h∆)hpyramid
hpyramid
h∆
b∆
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Instruction
Slide 158 / 159
Volume of a Sphere
A sphere is 2/3 the volume of a cylinder
with the same base area
(B = πr2) and height (h = d = 2r).
h=d
h = 2r
V = 2/3 (πr2h)
V = 2/3 πr2(2r)
V = 4/3πr3
V = 4/3π(6)3
V = 288π u3
V = 904.32 u3
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Instruction
Slide 159 / 159
Standards for Mathematical Practices
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.
MP8 Look for and express regularity in repeated reasoning.
Click on each standard to bring
you to an example of how to
meet this standard within the unit.