Chapter 11 Dimensional Geometry - Mangham Mathmanghammath.com/Chapter Packets/Chapter 11 Dimensional Geometry.pdf · Created by Lance Mangham, 6 th grade math, Carroll ISD ACCELERATED

Created by Lance Mangham, 6th grade math, Carroll ISD

ACCELERATED MATHEMATICS

CHAPTER 11

DIMENSIONAL GEOMETRY

TOPICS COVERED:

• Naming 3D shapes

• Nets

• Volume of Prisms

• Volume of Pyramids

• Surface Area of Prisms

• Surface Area of Pyramids

• Surface Area using Nets


Accelerated Mathematics Formula Chart Name:

Linear Equations

Slope-intercept form y mx b= + Direct Variation y kx= (8th grade)

Constant of proportionality y

kx

= Slope of a line 2 1

2 1

y ym

x x

−=

−

(8th grade)

Circumference Circle 2C rπ= or C dπ=

Area

Rectangle A bh= Trapezoid 1 2

1( )

2A b b h= +

Parallelogram A bh= Circle 2A rπ=

Triangle 2

bhA = or

1

2A bh=

Surface Area (8th grade) Lateral Total

Prism S Ph= 2S Ph B= +

Cylinder 2S rhπ= 22 2S rh rπ π= +

Volume

Triangular prism V Bh= Cylinder 2 or V Bh V r hπ= = (8th grade)

Rectangular prism V Bh= Cone 21 1

or 3 3

V Bh V r hπ= = (8th)

Pyramid 1

3V Bh= Sphere

34

3V rπ= (8th grade)

Pi 22

3.14 or 7

π π≈ ≈

Distance d rt= Compound Interest (1 )tA P r= +

Simple Interest I prt= Pythagorean Theorem 2 2 2a b c+ = (8th grade)

Customary – Length 1 mile = 1760 yards

1 yard = 3 feet

1 foot = 12 inches

Customary – Volume/Capacity 1 pint = 2 cups

1 cup = 8 fluid ounces

1 quart = 2 pints

1 gallon = 4 quarts

Customary – Mass/Weight 1 ton = 2,000 pounds

1 pound = 16 ounces

Metric – Length 1 kilometer = 1000 meters

1 meter = 100 centimeters

1 centimeter = 10 millimeters

Metric – Volume/Capacity 1 liter = 1000 milliliters

Metric – Mass/Weight 1 kilogram = 1000 grams

1 gram = 1000 milligrams


Area/Volume/Surface Area Computation Page

EXAMPLES

1 2

2

1( )

2

1(10 20) 6

2

90 cm

A b b h

A

A

= +

= + •

=

2

2

3

3.14 10 5

1570 m

V r h

V

V

π=

= • •

=

2

2

2(8 6) (28) 10

376 in

S B Ph

S

S

= +

= • + •

=


Cube Square prism

Rectangular prism Right triangular prism

Trapezoidal prism Isosceles triangular prism

Cylinder Cone

Triangular and Square Pyramids Sphere


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.


Activity 11-1: Basics of Solids Name:

Three – dimensional (solid) figures include what five shapes?

Characteristics of Solids

• Three-dimensional figures, or solids, can have ________________ or ________________ surfaces.

• Prisms and pyramids are named by the shapes of their ________________.

• A ________________ is a diagram of the surfaces of a three-dimensional figure. It can be folded to

form the three-dimensional figure.

Classify each solid and tell how many faces, edges, and vertices.

Type Properties

1. Faces = Edges = Vertices =





1 2 3 4 5


Activity 11-2: 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=

The volume V of a pyramid is one third the product of the area of the base, B and the height, h.

1

3V Bh=

Volume of a Prism

, Area of the baseV Bh B= =

Volume of a Pyramid

1, Area of the base

3V Bh B= =

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. 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?

7.

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?

8.

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?


Activity 11-3: Volume of Prisms Name:

Find the volume of the four prisms below.

Find the volume of the rectangular prism with length, l, width, w, and height, h.

5. = 5 m, 8 m, 9 ml w h= = 6. = 10 in, 14 in, 15 inl w h= =

7. = 16 yd, 10.2 yd, 4.3 ydl w h= = 8. 1

= 12 mm, 17 mm, 2 mm2

l w h= =

Find the volume of the solid. Round your answer to the nearest hundredth, if necessary.

7 in

7 in

7 in

13 mm

7 mm

4 mm

17 m

29 m

21 m

1. 2.

3. 4.

9.

10 ft

9 ft 2 ft

10.

11.


Activity 11-4: Volume of Pyramids Name:

Find the volume of each pyramid below.

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. 2 = 18 in , 5 inB h = 7. 2 = 6.3 mm , 2.9 mmB h =

Find the volume of the three pyramids below.

Find the volume of the square pyramid with base side length s and height h.

11. 3 in, 7 ins h= = 12. 9 mm, 14 mms h= = 13. 1

9 ft, ft2

s h= =

300 ft 300 ft

h = 321 ft

This is the Pyramid Arena in Memphis, TN.

13 cm 2 cm

7 cm

10 in 10 in

h = 15 in

8 m

11 m

Height of pyramid:

11 m

4 in

5 in

h = 8 in

1. 2.

8. 9. 10.


Activity 11-5: Volume of Prisms & Pyramids Name:

Find the volume of each solid below.

1. 2.

3. 4.

Solve.

5. A triangular prism has a base area of 20 square feet and a height of 4 feet. Find the volume.

6. The volume of a triangular pyramid is 300 cubic meters. What is the area of the pyramid’s base if

the pyramid height is 3 meters?

7. A triangular prism has a volume of 2,500 cubic feet. What is the length of the prism if its

triangular bases are right triangles, each with perpendicular sides of 10 and 20 feet?

8. The height of a pyramid is 15 inches. The pyramid’s base is a square with a side of 5 inches. What

is the pyramid’s volume?

9.

A rectangular box has a volume of 480 cubic inches. The height of the box 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?

10.

A square pyramid has edges of length p and a height of p as well. Which expression

represents the volume of the pyramid?

A 31

3p B

21

3p C

21

3p p+ D

21

3p p+

11.

Mr. Underwood says that when the height of a rectangular prism is doubled, its volume also

doubles. Mrs. Scogin says then when the height of a rectangular prism is doubled, volume

quadruples. Who is correct? Explain your reasoning.

12. What happens to the volume of a rectangular prism when its height is tripled?

13. What happens to the volume of a triangular pyramid when all dimensions are tripled?

14. What happens to the volume of a triangular prism when the area of its base is doubled?


Activity 11-6: 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. 2S Ph B= +

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. The triangles are equilateral triangles.

9 in

18 in

7.8 in



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

Add the area of all sides and base(s)S =

Lateral Surface Area

Add the area of all sidesS =

Surface Area of a Prism Total Surface Area

2 ,

Perimeter of base, Area of base

S Ph B

P B

= +

= =

Lateral Surface Area

Perimeter of base

S Ph

P

=

=

Please measure to the nearest tenth of a centimeter.

Dimensions: ________, _________, ________

LS = ________________

S = _________________



Find the total and lateral surface area of each figure. Assume the top and bottom faces are the bases.

Don’t forget to include units!

1. 2.

3. 4.

5. A glass, equilateral 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?

Draw a net, find the area of each face, and find the total of all the areas

6. 7.

12 in

11 in

10 in

8 m

1 m

3 m



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.

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.

3. 28 m , 3 m, 6 mB P h= = = 4. 215 m , 12 m, 3 mB P h= = =

5. 242 yd , 23 yd, 8 ydB P h= = = 6. 258 mm , 36 mm, 20 mmB P h= = =

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.

11. 12.

10 in

5 in

3 in

13 ft

6 ft

4 ft

17 in

18 in 2 in


Activity 11-10: Surface Area of 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

base perimeter P and the slant height l. 1

2S B Pl= +

What is the height of the pyramid? _____

What is the slant height of the pyramid? _____

What is the base shape? ____________

What does B stand for? ____________

What does P stand for? ____________

LSA = _________

TSA = __________

V = ___________

20 cm

20 cm

24

26

h cm

l cm

=

=


Activity 11-11: Surface Area of Pyramids Name:

Find the total surface area of the four regular pyramids shown by drawing a net first. Then find the

lateral surface area of each.

1. 2.

3. 4.

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 rectangular pyramid with the base lengths shown and the slant height, l.

6. length = width 4 m= , 13 ml = 7. length = 6 in, width 10 in= , 15 inl =

8. length = width 11 m= , 17.3 ml = 9. length = 16 yd, width 20 yd= , 14.1 ydl =

10. 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?

7 ft 7 ft

6 ft

5 ft 5 ft

6 ft

4 congruent faces


Activity 11-12: Surface Area of Prisms & Pyramids Name:

1. What is the total surface area of this cube?

A 21

91 ft8

B 21

20 ft4

C 21

121 ft2

D 21

101 ft4

Find the surface area of the following shapes.

2. 3.

Draw a net and find the area of each face to calculate the lateral and total surface areas for each figure.

4. The base is a square. 5. 6.

7. 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 gift box. Find

the surface area of the box.

Length of gift paper:__________ Width of gift paper:__________ Surface Area:__________


Activity 11-13: 3D Shapes/Volume/Surface Area Name:

1.

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?

2.

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?

3. How many 2 by 2 by 2 inch cubes will fit into a 4 by 8 by 12 inch box?

4.

For a regular pentagonal prism, what is the ratio of the number of vertices

to the number of edges?

2:3 3:2 3:5 5:3

(43% of all 11th graders answered this question correctly on STAAR.)

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.

6.

Identify the three-dimensional figure that can be formed by this net:

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.

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.

9. How many cubic inches are in a cubic foot?


Activity 11-14: Volume and Capacity Name:

Taken from TexTEAMS Rethinking Elementary Math, Part II

• 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 Total number of cubes Hidden Cubes

1 1 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.

• 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 Total number of cubes

2 by 3 by 4 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?

Documents

Chapter 11 Dimensional Geometry - Mangham Mathmanghammath.com/Chapter Packets/Chapter 11 Dimensional Geometry.pdf · Created by Lance Mangham, 6 th grade math, Carroll ISD ACCELERATED