ACCELERATED MATHEMATICS
CHAPTER 11
DIMENSIONAL GEOMETRY
TOPICS COVERED:
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Naming 3D shapes
Nets
Volume of Prisms
Volume of Pyramids
Surface Area of Prisms
Surface Area of Pyramids
Surface Area using Nets
Created by Lance Mangham, 6th grade math, Carroll ISD
Accelerated Mathematics Formula Chart
Name:
Linear Equations
Slope-intercept form
y = mx + b
Constant of proportionality
k=
Circumference
y
x
Direct Variation
y = kx (8th grade)
Slope of a line
m=
y2 − y1
(8th grade)
x2 − x1
C = 2π r or C = π d
Circle
Area
1
(b1 + b2 ) h
2
Rectangle
A = bh
Trapezoid
A=
Parallelogram
A = bh
Circle
A = π r2
Triangle
A=
bh
1
or A = bh
2
2
Surface Area (8th grade)
Prism
Cylinder
Lateral
Total
S = Ph
S = Ph + 2 B
S = 2π rh
S = 2π rh + 2π r 2
Volume
Triangular prism
V = Bh
Cylinder
Rectangular prism
V = Bh
Cone
Pyramid
1
V = Bh
3
Sphere
V = π r 2 h or V = Bh (8th grade)
1
1
V = Bh or π r 2 (8th grade)
3
3
4
V = π r 3 (8th grade)
3
22
7
Pi
π ≈ 3.14 or π ≈
Distance
d = rt
Compound Interest
A = P (1 + r )t
Simple Interest
I = prt
Pythagorean Theorem
a 2 + b 2 = c 2 (8th grade)
Customary – Length
1 mile = 1760 yards
1 yard = 3 feet
1 foot = 12 inches
Metric – Length
1 kilometer = 1000 meters
1 meter = 100 centimeters
1 centimeter = 10 millimeters
Customary – Volume/Capacity
1 pint = 2 cups
1 cup = 8 fluid ounces
1 quart = 2 pints
1 gallon = 4 quarts
Metric – Volume/Capacity
1 liter = 1000 milliliters
Customary – Mass/Weight
1 ton = 2,000 pounds
1 pound = 16 ounces
Metric – Mass/Weight
1 kilogram = 1000 grams
1 gram = 1000 milligrams
Created by Lance Mangham, 6th grade math, Carroll ISD
Area/Volume/Surface Area Computation Page
EXAMPLES
1
A = (b1 + b2 )h
2
1
A = (10 + 20) • 6
2
A = 90 cm 2
V = π r 2h
S = 2 B + Ph
2
V = 3.14 • 10 • 5
V = 1570 m
3
S = 2(8 • 6) + (28) • 10
S = 376 in 2
Created by Lance Mangham, 6th grade math, Carroll ISD
Cube
Square prism
Rectangular prism
Right triangular prism
Trapezoidal prism
Isosceles triangular prism
Cylinder
Cone
Triangular and Square Pyramids
Sphere
Created by Lance Mangham, 6th grade math, Carroll ISD
Net of a pyramid
Net of a cylinder
Lateral face: A face that joins the
bases of a solid. It is any edge or face
that is not part of the base.
Created by Lance Mangham, 6th grade math, Carroll ISD
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-2: Examples of Solids
Name:
Classify each solid and tell how many faces, edges, and vertices.
Type
Picture of Solid
Properties
1.
How many of each?
• Faces
• Edges
• Vertices
2.
How many of each?
• Faces
• Edges
• Vertices
3.
How many of each?
• Faces
• Edges
• Vertices
4.
How many of each?
• Faces
• Edges
• Vertices
5.
How many of each?
• Faces
• Edges
• Vertices
6.
How many of each?
• Faces - 2
• Edges - 0
• Vertices - 0
7.
How many of each?
• Faces - 1
• Edges - 0
• Vertices – 0 (1 apex)
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-3: Volume of Prisms & Pyramids
Name:
The volume of a solid is a measure of the amount of space it occupies or how much it can hold.
The volume V of a prism is the product of the area of the base B and the height h. V = Bh
1
The volume of a pyramid is one third the product of the area of the base, B and the height, h. V = Bh
3
Volume of a Prism
Volume of a Pyramid
1
V = Bh, B = Area of the base
3
V = Bh, B = Area of the base
Find the volume of each figure.
1.
2.
Identify the three-dimensional shape that can be formed from each net.
3.
4.
5.
Solve.
6.
7.
8.
The base of a rectangular pyramid is 13 inches long and 12 inches wide.
The height of the pyramid is 8 inches. What is the volume of the pyramid?
A cake pan is shaped like a rectangular prism. The pan’s volume is 216
in3. The cake pan has a base that is 12 inches by 9 inches. What is the
height of the cake pan?
A form for a garden ornament is made up of two shapes, a cube and a
square pyramid (see picture at the right above this table). To make an
ornament the form is filled with concrete. What is the volume of the
form?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-4: Volume of Prisms & Pyramids
Name:
Find the volume of each solid below.
1.
2.
_________________________________
3.
__________________________________
4.
_________________________________
__________________________________
Solve.
A triangular prism has a base area of 20 square feet and a height of 4 feet.
5.
Find the volume.
The volume of a triangular pyramid is 300 cubic meters. What is the area of
6.
the pyramid’s base if the pyramid height is 3 meters?
A triangular prism has a volume of 2,500 cubic feet. What is the length of
7. the prism if its triangular bases are right triangles, each with perpendicular
sides of 10 and 20 feet?
The height of a pyramid is 15 inches. The pyramid’s base is a square with a
8.
side of 5 inches. What is the pyramid’s volume?
A rectangular box has a volume of 480 cubic inches. The height of the box
9. is 5 inches. The ratio of the length of the box to the width of the box is 3 to
2. What is the measure of the width of the box?
A square pyramid has edges of length p and a height of p as well. Which
expression represents the volume of the pyramid?
10.
1
1
1 2
1
A p3
B p2
C
D p + p2
p +p
3
3
3
3
11. Find the volume of the rectangular prism.
7
V = ___________
12
12
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-5: Volume of Prisms & Pyramids
Name:
Find the volume of the four prisms below.
9 ft
2 ft
10 ft
7 in
7 in
7 in
Find the volume of the rectangular prism with length, l, width, w, and height, h.
5. l = 5 m, w = 8 m, h = 9 m
6. l = 10 in, w = 14 in, h = 15 in
7. l = 16 yd, w = 10.2 yd, h = 4.3 yd
8. l = 12 mm, w = 17 mm, h = 2
1
mm
2
Find the volume of the solid. If two units of measure are used, give your answer in the smaller units.
Round your answer to the nearest hundredth.
7 mm
13 mm
17 m
21 m
4 mm
29 m
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-6: Volume of Prisms & Pyramids
Name:
Find the volume of each pyramid below.
h = 321 ft
8 in
5 in
300 ft
300 ft
4 in
This is the Pyramid Arena in Memphis, TN.
Find the volume of the pyramid.
3. Triangular pyramid: base of the triangle = 8 ft, height of the triangle = 6 ft, height of the pyramid = 7
ft
4. Square pyramid: sides of the square = 14 mm, height of the pyramid = 9 mm
5. A pyramid bookend is being formed out of concrete. The rectangular base on the bookend is 8 in by
7 in. The height of the pyramid is 5 in.
Find the volume of the pyramid with base area B and height h.
6. B = 18 in 2 , h = 5 in
7. B = 6.3 mm 2 , h = 2.9 mm
Find the volume of the three pyramids below.
h = 15 in
Height of pyramid:
11 m
7 cm
13 cm
11 m
2 cm
10 in
10 in
8m
Find the volume of the square pyramid with base side length s and height h.
11. s = 3 in, h = 7 in
12. s = 9 mm, h = 14 mm
13. s = 9 ft, h =
1
ft
2
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-7: Surface Area of Prisms
Name:
A net is a two-dimensional pattern that forms a solid when it is folded.
The surface area of a polyhedron is the sum of the areas of its faces.
The total surface area of a prism is the sum of twice the area of the base and the product of the base’s
perimeter and the height. S = 2 B + Ph
Example
Draw a net for the pentagonal prism. For the rectangular faces, draw adjacent rectangles. Draw the
bases on opposite sides one rectangle.
Draw a net for the following shapes.
A lamp shade will be constructed from rice paper shown below. How much paper will be needed to
make the lampshade? The first time, use the sum of the areas method. The second time, use the formula
for the surface area of a prism.
9 in
18 in
7.8 in
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-8: Surface Area of Prisms
Name:
The total surface area of a solid is the sum of the areas of all faces including the bases.
The lateral surface area of a solid is the sum of all faces excluding the bases. In a rectangular prism, you
can assume the bases are the top and bottom faces, unless otherwise specified.
Surface Area from a Net of a Prism or Pyramid
Total Surface Area
Lateral Surface Area
SA = Add the area of all the sides and base
SA = Add the area of all the sides
Surface Area of a Prism
Total Surface Area
Lateral Surface Area
SA = Ph + 2 B,
SA = Ph
P = Perimeter of base
P = Perimeter of base, B = Area of base
Rectangular Prism
SA = 2lw + 2lh + 2wh
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-9: Surface Area of Prisms
Name:
Find the surface area of each figure. Don’t forget to include units!
1.
2.
3.
4.
5.
The top of the Washington Monument is a triangular pyramid with a square base. Each triangular
face is 58 feet tall and 34 feet wide and covered with white marble. About how many square feet of
marble cover the faces of the pyramid?
6.
A glass triangular prism for a telescope is 5.5 inches long. Each side of the prism’s triangular bases
is 4 inches long and 3 inches high. How much glass covers the surface of the prism?
4 ft
7. Michael is refinishing the bookcase pictured to the left. A pint of stain
covers 30 – 35 ft2. How many cans of stain will Michael need to buy to cover
the left side, right side, and back of the book case with two coats?
6 ft
6 in.
8. The Imaginary Toy Company has increased their size of the “Creativity Doll”.
The packaging department has calculated that they need to add 3 inches to each of
the dimension of the original packaging. What is the new amount of cardboard
needed to package one doll?
10 in
5 in
3 in
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-10: Surface Area of Prisms
Name:
1. You are making a jewelry box shown for your mother. Draw a net and then find the amount of wood
you need to make the box. Then use the formula for the surface area of a prism.
3 in
5 in
10 in
2. What is the surface area of a rectangular prism that is 6 inches long, 8 inches wide, and 2 inches
high?
Find the surface area of the prism, where B is the area of the base, P is the perimeter of the base, and h is
the height.
4. B = 15 m 2 , P = 12 m, h = 3 m
3. B = 8 m 2 , P = 3 m, h = 6 m
5. B = 42 yd 2 , P = 23 yd, h = 8 yd
6. B = 58 mm 2 , P = 36 mm, h = 20 m
Identify the solid shown by the net. Then find the surface area.
7.
8.
Draw a net for the solid. Then find the surface area of the solid.
9.
10.
6 ft
13 ft
11.
4 ft
12.
18 in
2 in
17 in
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-11: Surface Area of Prisms & Pyramids
Name:
The slant height, l, of a regular pyramid is the height of any of its triangular faces.
The surface area S of a pyramid is the sum of the area of the base B and one half the product of the
1
base perimeter P and the slant height l. S = B + Pl
2
What is the height of the pyramid? _____
h = 24 cm
l = 26 cm
What is the slant height of the pyramid? _____
LSA = _________
SA = __________
What is the base shape? ____________
V = ___________
What does B stand for? ____________
What does P stand for? ____________
20 cm
20 cm
Please measure to the nearest tenth of a centimeter.
Dimensions: ________, _________, ________
LSA = ________________
SA = _________________
Find the surface area of each net.
1. Each square is one square meter.
2. Each square is one square yard
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-12: Surface Area of Pyramids
Name:
Find the surface area of the four regular pyramids shown by drawing a net first.
6 ft
7 ft
7 ft
8m
6 ft
5 ft
5 ft
5 in
4m
4m
7 in
7 in
5. Find the surface area of a pyramid whose slant height is 9 cm and whose base is a 4 cm by 6 cm
rectangle.
Find the surface area of the pyramid with the base lengths shown and the slant height, l.
6. length = width = 4 m , l = 13 m
7. length = 6 in, width = 10 in , l = 15 in
8. length = width = 11 m , l = 17.3 m
9. length = 16 yd, width = 20 yd , l = 14.1 yd
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-13: Surface Area of Prisms & Pyramids
Name:
Draw a net, find the area of each face, and find the total of all the areas
1.
2.
8m
12 in
1m
11 in
3m
10 in
3. A rectangular prism 1ft by 7 ft by 8 ft.
4. A rectangular prism 3.5 ft by 4.5 ft by 6 ft.
5. You are wrapping a gift box that is 15 inches long, 12 inches wide, and 4 inches deep. Use a net to
find the length and width of a single sheet of paper that could be used to wrap the entire girft box. Find
the surface area of the box.
Length of gift paper:__________ Width of gift paper:__________ Surface Area:__________
Find the surface area of each shape. Remember to label!!!
Draw a net, find the area of each face, and find the total of all the areas.
6.
7.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-14: Volume and Surface Area
Name:
1. What is the total surface area of this cube?
A 91
1 2
ft
8
B 20
1 2
ft
4
C 121
1 2
ft
2
D 101
1 2
ft
4
2.
Izabella takes her pet iguana to the pet store. The owner of the store tells her that she needs to
buy a new terrarium that has twice as much volume as the old one. Which of the following
processes should she use to determine the dimensions of the new terrarium that has twice the
volume of her current one?
A She should double one dimension.
D She should double two dimensions.
B She should double all three dimensions.
E She should square one dimension.
C She should cube one dimension.
3.
How many cubic inches are in a cubic foot?
Find the surface area of the following shapes.
4.
5.
Calculate the lateral and total surface areas for each figure.
6. The base is a square.
7.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-15: 3D Shapes/Volume/Surface Area
1.
2.
3.
Name:
A shipping company sells two types of cartons that are shaped like
rectangular prisms. The larger carton has a volume of 720 cubic inches.
The smaller carton has dimensions that are half the size of the larger
carton. What is the volume, in cubic inches, of the smaller carton?
An ice cream carton has a volume of 64 fluid ounces. A second ice cream
carton has dimensions that are three-fourths the size of the larger carton.
What is the volume of the smaller carton?
How many 2 by 2 by 2 inch cubes will fit into a 4 by 8 by 12 inch box?
For a regular pentagonal prism, what is the ratio of the number of vertices
to the number of edges?
4.
2:3
3:2
3:5
5:3
(43% of all 11th graders answered this question correctly on TAKS.)
5.
Name the solid that has 2 bases that are each 5 sided shapes and the
vertices of each base are joined together forming 5 edges.
Identify the three-dimensional figure that can be formed by this net:
6.
7.
You have two cubes. The smaller cube has dimensions that are x long
and the larger cube has dimensions that are 4x long. If the smaller cube
has a volume of 64 cubic feet, find the volume of the larger cube.
8.
Identify the following solids:
A. I have six flat faces. I have twelve edges. I have eight vertices.
B. I have five flat faces. I have nine edges. I have six vertices.
C. I have two flat faces. I can roll.
D. I have five flat faces. I have eight edges. I have five vertices.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 11-16: Volume and Capacity
Name:
Taken from TexTEAMS Rethinking Elementary Math, Part II
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If the first cube is 1 unit on an edge, it takes 1 unit cube to build it.
The next larger cube is 2 units on an edge. It takes 8 units cubes to build it. Notice that there is
one cube hidden by the other cubes no matter how hard you look for it.
Continuing on with the next cubes, record the number of unit cubes it takes to build each and the
number of hidden cubes.
Number of cubes on each edge
1
Total number of cubes
1
Hidden Cubes
0
2
8
1
3
4
5
6
1. If you know the number of units on the edge of a cube, explain how you could find out the number
of cubes it would take to build.
2. If you know the number of unit cubes it takes to build a larger cube, can you figure out the units on
an edge?
3. Do you notice any pattern about the hidden number of cubes? If so, describe it.
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Use the cubes to build a variety of rectangular prisms.
For each one record the dimensions of your prism and the number of cubes it took to build it.
Dimensions of prism
2 by 3 by 4
Total number of cubes
12
4. Describe the patterns you see in your data.
5. Can you figure out the dimensions of your prism if you know the number of cubes it took to build it?
6. Can you figure out how many cubes it took to build your prism if you know its dimensions?
Created by Lance Mangham, 6th grade math, Carroll ISD