7th Grade Math 3D Geometry
2017­02­13
www.njctl.org
Table of Contents
3­Dimensional Solids
Cross Sections of 3­Dimensional Figures
Volume
• Prisms and Cylinders
• Pyramids, Cones & Spheres
Surface Area
•
•
•
•
Prisms
Pyramids
Cylinders
Spheres
More Practice/ Review
Glossary & Standards
Teacher Notes
Click on a topic to go to that section
Vocabu
in the p
box the
linked to
of the p
word de
3­Dimensional Solids
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Click below for a website with interactive 3D shapes and nets
Polyhedron
A polyhedron is a 3­D figure whose faces are all polygons.
Sort the figures to the appropriate side.
Polyhedron
Not Polyhedron
3­Dimensional Solids
Categories & Characteristics of 3­D 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
3­Dimensional Solids
Categories & Characteristics of 3­D Solids:
Cylinders
1. Have 2 congruent, circular bases which are parallel to one another
click to reveal
2. Sides are curved
Cones
1. Have 1 circular base with a vertex opposite it
2. Sides are curvedclick to reveal
3­Dimensional Solids
Vocabulary Words for 3­D Solids:
Polyhedron A 3­D 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)
Name the figure. A
Rectangular Prism B
Triangular Pyramid C
Hexagonal Prism D
Rectangular Pyramid
E
Cylinder F
Cone Answer
1
Name the figure. A
B
C
D
E
F
Rectangular Pyramid Triangular Prism Octagonal Prism Circular Pyramid Cylinder Cone Answer
2
Name the figure. A
B
C
D
E
F
Rectangular Pyramid Triangular Pyramid Triangular Prism Hexagonal Pyramid Cylinder Cone Answer
3
Name the figure. A
B
C
D
E
F
Rectangular Prism Triangular Prism Square Prism Rectangular Pyramid Cylinder Cone Answer
4
Name the figure. A
B
C
D
E
F
Rectangular Prism Triangular Pyramid Circular Prism Circular Pyramid Cylinder Cone Answer
5
Faces, Vertices and Edges
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Click
Math Practice
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?
Euler's Formula
Euler's Formula: E + 2 = F + V
click to reveal
The sum of the edges and 2 is equal to the sum of the faces and vertices.
How many faces does a pentagonal prism have?
Answer
6
How many edges does a rectangular pyramid have?
Answer
7
How many vertices does a triangular prism have?
Answer
8
Cross Sections of Three­Dimensional Figures
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Cross Sections
3­Dimensional figures can be cut by planes. When you cut a 3­D figure by a plane, the result is a 2­D figure, called a cross section. These cross sections of 3­D figures are 2 dimensional figures you are familiar with.
Look at the example on the next page to help your understanding.
Cross Sections
A horizontal cross­section of a cone is a circle.
Can you describe a vertical cross­section of a cone?
Cross Sections
A vertical cross­section of a cone is a triangle.
Cross Sections
A water tower is built in the shape of a cylinder.
How does the horizontal cross­section compare to the vertical cross­section?
Cross Sections
The horizontal cross­section is a circle.
The vertical cross­section is a rectangle
Which figure has the same horizontal and vertical cross­sections?
A
C
B
D
Answer
9
10 Which figure does not have a triangle as one of its cross­sections?
C
B
D
Answer
A
A
Triangle B
Circle C
Rectangle
D
Trapezoid Answer
11 Which is the vertical cross­section of the figure shown?
A
Triangle B
Circle C
Rectangle D
Trapezoid Answer
12 Which is the horizontal cross­section of the figure shown?
A
Triangle B
Circle C
Square D
Trapezoid Answer
13 Which is the vertical cross­section of the figure shown?
14 Misha has a cube and a right­square pyramid that are made of clay. She placed both clay figures on a flat surface.
Select each choice that identifies the two­dimensional plane sections that could result from a vertical or horizontal slice through each clay figure.
B Cube cross section is a Square
C Cube cross section is a Rectangle (not a square)
D Right­Square Pyramid cross section is a Triangle
E Right­Square Pyramid cross section is a Square
F Right­Square Pyramid cross section is a Rectangle (not a square)
From PARCC EOY sample test calculator #11 Answer
A Cube cross section is a Triangle
Note: cross s
you w
Volume
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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
Units3 or cubic units
Math Practice
Label
MP6: At
Continu
units us
answer
Ask: Wh
use?
Volume Activity
Lab #1: Volume Activity
Math Practice
Click the link below for the activity
Volume of Prisms & Cylinders
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Volume
Volume of Prisms & Cylinders:
Click
Area of Base x Height
Area Formulas:
Rectangle = lw or bh
Click
Click
Triangle = or Circle = r2
Click
Volume
8 m
2 m
5 m
Answer
Find the Volume.
Volume
Answer
Find the Volume. Use 3.14 as your value of .
9 yd
10 yd
Answer
15 Find the volume.
Answer
16 Find the volume of a rectangular prism with length 2 cm, width 3.3 cm and height 5.1 cm.
A
B
C
D
Answer
17 Which is a possible length, width and height for a rectangular prism whose volume = 18 cm 3?
18 Find the volume.
50 ft
42 ft
21 ft
Answer
47 ft
19 Find the volume. Use 3.14 as your value of .
10 m
Answer
6 m
Teachers:
Use this Mathematical Practice Pull Tab for the next 3 SMART Response slides.
Math Practice
The nex
MP4 & Ask: W
see bet
prism/c
(MP4)
Could y
your th
Why do
(MP4)
20 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?
Answer
HINT: You may want to draw a picture!
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 .
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 . Teachers:
Use this Mathematical Practice Pull Tab for the next SMART Response slide.
Math Practice
The ne
Ask: W
will he
W
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
Answer
23 Which circular glass holds more water? Use 3.14 as your value of . Before revealing your answer, make sure that you can prove that your answer is correct.
Volume of Pyramids, Cones & Spheres
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Volume
Math Practice
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?
Comparing Volume of 3D Shapes
Click here for a demonstration comparing volume of cones & spheres with volume of cylinders
Volume of a Cone
The Volume of a Cone is the volume of a cylinder with the same base area (B) and height (h).
What is the formula?
Volume of a Sphere
The Volume of a Sphere is the volume of a cylinder with the same base area (B) and height (h).
What is the formula?
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?
Answer
(Just Ice Cream within Cone. Not on Top)
9 in
4 in
Answer
24 Find the volume.
25 Find the volume.
5 cm
Answer
8 cm
Volume
If the radius of a sphere is 5.5 cm, what is its volume?
Answer
26 What is the volume of a sphere with a radius of 8 ft?
Answer
27 What is the volume of a sphere with a diameter of 4.25 in?
Volume of a Pyramid
The Volume of a Pyramid is the volume of a prism with the same base area (B) and height (h).
What is the formula?
Note: Pyramids are named by the shape of their base.
Volume
Example: Find the volume of the pyramid shown below.
=5 m
m
4 =
le
e
sid
n
h t
g
Answer
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
29 Find the volume.
7 cm
8 cm
Answer
15.3 cm
Surface Area
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Surface Area of Prisms
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Surface Area
Surface Area is the sum of the areas of all outside surfaces of a 3­D 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
Surface Area
6 in
3 in
8 in
8 in
3 in
6 in
6 in
6 in
8 in
3 in
3 in
8 in
Bottom: 8 x 3 = 24
Left: 6 x 3 = 18
Front: 8 x 6 = 48
Top: 8 X 3 = 24
Right: 6 X 3 = 18
Back: 8 x 6 = 48
Sum: 24 + 24 + 18 + 18 + 48 + 48 = 180 in2
Surface Area: 180 in2
Net
Another way that you can visualize the entire surface and calculate the surface area is to create the net of your solid by unfolding it.
Below is a the net of a rectangular prism. Back
Left
Bottom
Front
Right
Top
Surface Area using Nets
You can also calculate the surface area of our last example by drawing the net, calculating the areas, and adding them together.
48 in2
6 in
18 in2
24 in2
18 in2
24 in2
3 in
8 in
48 in2
Total Surface Area: 18 + 24 + 18 + 24 + 48 + 48 = 180 in2 Arrangement of Unit Cubes
Surface Area Activity
Click the link below for the activity
Lab #2: Surface Area Activity
The ne
Use this Mathematical Practice Pull Tab for the next 4 SMART Response slides.
Math Practice
Teachers:
Ask: Ho
the pro
Wha
answ
A
B
C
Answer
30 Which arrangement of 27 cubes has the least surface area?
31 Which arrangement of 12 cubes has the least surface area?
Answer
A
B
C
D
A
B
Answer
32 Which arrangement of 25 cubes has the greatest surface area?
A
B
C
D
E
F
Answer
33 Which arrangement of 48 cubes has the least surface area?
Surface Area
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.
5 in
6 in
12 in
Top/Bottom
Front/Back
Left/Right
Surface Area
12
x 6
72
x 2
144
12
x 5
60
x 2
120
6
x 5
30
x 2
60
144
+ 120
324 in2
click to reveal
click to reveal
click to reveal
click to reveal
Surface Area
Name the figure.
Find the figure's surface area.
Answer
Re
7 m
Base 3(5)(2) 30 Sum = 14
3 m
5 m
Click to reveal the prism's net, if needed
34 How many faces does the figure have?
Answer
6 m
2 m
4 m
Answer
35 How many area problems must you complete when finding the surface area?
6 m
2 m
4 m
3
36 What is the area of the top or bottom face?
Answer
6 m
2 m
4 m
37 What is the area of the left or right face?
Answer
6 m
2 m
4 m
38 What is the area of the front or back face?
Answer
6 m
2 m
4 m
39 What is the surface area of the figure?
Answer
6 m
2 m
4 m
Click to reveal the prism's net, if needed
2
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
5 yd
5 yd
4 yd
9 yd
6 yd
Teacher Notes
Find the Surface Area
Surface Area
5 yd
5 yd
5 yd
4 yd
5 yd
9 yd
6 yd
Bottom Rectangle
9 Click
x 6
54
4 yd
5 yd
5 yd
4 yd
9 yd
9 yd
6 yd
6 yd
Triangles Left/Right Rectangles
Click
4
5 Click
x 6
x 9
24 / 2 = 12 45
x 2
x 2
24
90
Total Surface Area: 54 + 24 + 90 = 168 yd
2
Click
Find the Surface Area Using the Net
5 yd
5 yd
4 yd
9 yd
6 yd
Triangles
Click
4 x 6
24 / 2 = 12
x 2
24
Middle Rectangle
Left/Right Rectangles
9 Click
(Same size since isosceles)
x 6
5 x 9 = 45 x 2 = 90 Click
54
Total Surface Area: 24 + 54 + 90 = 168 ydClick2
Find the Surface Area
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
9 cm
9 cm
7.8 cm
9 cm
11 cm
Surface Area
9 cm
9 cm
7.8 cm
9 cm
11 cm
7.8 cm
11 cm
9 cm
Triangles Rectangles
Click
Click
Hint & Answer
9 cm
9 cm
40 Find the surface area of the shape below.
1. Draw
use t
2. Find t
50 ft
21 ft
42 ft
Hint & Answer
47 ft
3. Ca
4. Find
5
Find the Surface Area
3 cm
9 cm
4 cm
15 cm
5 cm
6 cm
Find the Surface Area
3 cm
9 cm
4 cm
15 cm
Answer
5 cm
6 cm
Trapezoids
Click 12
+ 6 18
x 4 72 / 2 = 36
x 2
72
Bottom Rectangle
Click
6
x 15
90
Top Rectangle
Click
12
x 15
180
Side Rectangles
Click
5
x 15 75
x 2 150
72 + 90
41 Find the surface area of the shape below.
8 cm
9 cm
10 cm
Answer
6 cm
42 Find the surface area of the shape below.
2 cm
cm
10 10 cm
m
6 c
Answer
6 cm
Surface Area of Pyramids
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Surface Area of Pyramids
What is a pyramid?
Click
Polyhedron with one base and triangular faces that meet at a vertex
How do you find Surface Area?
Click
Sum of the areas of all the surfaces of a 3­D f
igure
Find the Surface Area
17.4 cm
8 cm
7 cm
Teacher Notes
17.5 cm
Bottom Rectangle
Front/Back
Triangles
Left/Right
Triangles
Answer
Find the Surface Area.
56 + 13
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 x 4 = 16
Click
Click
4 Triangles:
4 in
Surface Area: 16 + 24 = 40 in2
Click
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 equivalent!).
Click
Remaining Triangles (all equal)
Hint
Click
Base
6 in
4 in
3.5 in
4 in
4 in
Click
Surface Area
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?
Square Pyramid
Cube
Answer
A
B
10 in
8 in
6.9 in
8 in
8 in
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.
11 m
12 m
6.7 m
9 m
9 m
Answer
45 Find the Surface Area.
Surface Area of Cylinders
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Surface Area
How would you find the surface area of a cylinder?
Hint
Surface Area
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.
Surface Area
Radius
Radius
Radius
H
E
I
G
H
T
Original cylinder
Curved Side = Circumference of Circular Base
Middle step to get to the net
H
E
I
G
H
T
Curved Side = Circumference of Circular Base
Net of a cylinder
Surface Area
Radius
Radius
H
E
I
G
H
T
H
E
I
G
H
T
Curved Side = Circumference of Circular Base
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 .
Area of Circles:
Click
16 in
14 in
Area of Curved Surface = Circumference Height
Click
Surface Area:Click
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 .
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 .
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 .
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 .
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 .
No
Answer
Yes Surface Area of Spheres
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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 .
Surface Area of a Sphere
Radius
click to reveal
Surface Area
If the diameter of the Earth is 12,742 km, what is its surface area? Use 3.14 as your value of . Round your answer to the nearest whole number.
12,742 km
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 . Click
2.7 in
Answer
51 Find the surface area of a softball with a diameter 3.8 inches. Use 3.14 as your value of . 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 . 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 . More Practice / Review
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54 Find the volume.
8 mm
15 mm
Answer
22 mm
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!
Answer
56 Find the volume of a square pyramid with base edge of 4 inches and pyramid height of 3 inches.
Answer
57 Find the Volume.
11 m
12 m
6 m
9 m
9 m
21 ft
14 ft
Answer
58 Find the Volume. Use 3.14 as your value of .
8 in
6.9 in
Answer
59 Find the Volume. Use 3.14 as your value of .
60 Find the volume.
Answer
8.06 ft
8 ft
4 ft
7 ft
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 .
Answer
20 cm
14 cm
25 cm
62 Find the surface area of the cylinder. Use 3.14 as your value of .
Answer
Radius = 6 cm and Height = 7 cm
63 Find the Surface Area.
11 cm
12 cm
Answer
11 cm
64 Find the Surface Area.
9 in
8.3 in
8 in
7 in
Answer
9 in
65 Find the volume.
9 in
8.3 in
8 in
7 in
Answer
9 in
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?
Answer
67 Find the surface area of a square pyramid with a base length of 4 inches and slant height of 5 inches.
68 Find the volume.
40 m
80 m
Answer
70 m
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?
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Teacher Notes
Glossary & Standards
Vocabu
in the p
box the
linked t
of the p
word de
Cone
A 3­D solid that has 1 circular base with a vertex opposite it.
The sides are curved.
Back to Instruction
Cross Section
The shape formed when cutting straight through an object. Triangle
Square
Trapezoid
Back to Instruction
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
Face
Flat surface of a Polyhedron
1 face
1 face
1 face
Back to Instruction
Edge
Line segment formed where 2 faces meet.
1 edge
1 edge
1 edge
Back to Instruction
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:
vertices = 4
faces = 4
E + 2 = F + V
E + 2 = 4 + 4
E + 2 = 8
E = 6
Back to Instruction
Net
A 2­D pattern of a 3­D solid that can be folded to form the figure. An unfolded geometric solid.
Solid
Solid
Net
Net
Solid
Net
Back to Instruction
Polyhedron
A 3­D figure whose faces are all polygons.
A Polyhedron has NO curved surfaces.
Plural: Polyhedra
Yes, polyhedron
Yes, polyhedron
No, not a polyhedron
Back to Instruction
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
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
Surface Area
The sum of the areas of all outside surfaces of a 3­D figure.
5
3
4
6
3 3 6 u25
5
4
6
24 u2 30 u2 6
4
5
3 6 u25
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 Back to Instruction
Vertex
Point where 3 or more faces/edges meet
Plural: Vertices
1 vertex
1 vertex
1 vertex
Back to Instruction
Volume
The amount of space occupied by a 3­D Figure. The number of cubic units needed to FILL a 3­D Figure (layering).
Label: V = ?
cylinder filled halfway with water
V = ?
Prism filled with water
Units3 or cubic units
Back to Instruction
Volume of a Cone
A cone is 1/3 the volume of a cylinder with the same base area (B = r2) and height (h).
V = ( r2h) ÷ 3
or
1 2h
V = r
3
h
6
4
r
1 2h
V = r
3
V = 1/3 (4)2(6)
V = 32 units3 V = 100.48 u3 Back to Instruction
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 = r2h
4
r
h
V = r2h
10
V = (4)2(10)
V = 160 units3 V = 502.4 u3 Back to Instruction
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
Volume of a Pyramid
A pyramid is 1/3 the volume of a prism with the same base area (B) and height (h).
1 V = (Bh) ÷ 3 or V = (Bh)
3
The shape of your base matches the name of the pyramid
Rectangular
Pyramid
V = 1/3Bh
V = 1/3(lw)h
Triangular
Pyramid
V = 1/3Bh
V = (1/2b∆h∆)hpyramid
hpyramid
h∆
b∆
Back to Instruction
Volume of a Sphere
A sphere is 2/3 the volume of a cylinder with the same base area (B = r2) and height (h = d = 2r).
6
h = d h = 2r
V = 2/3 ( r2h)
V = 2/3 r2(2r)
V = 4/3 r3
V = 4/3 (6)3 V = 288 u3
V = 904.32 u3
Back to Instruction
Standards for Mathematical Practices
Throughout this unit, the Standards for Mathematical Practice are used.
MP1: Making sense of problems & persevere in solving them.
MP2: Reason abstractly & 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 & make use of structure.
MP8: Look for & express regularity in repeated reasoning.
Additional questions are included on the slides using the "Math Practice" Pull­tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull­tab.