Surface Areas of Solids - Mrs. Middleton Algebra & Geometry

Surface Areas of Solids6.1 Drawing 3-Dimensional Figures6.1 Drawing 3-Dimensional Figures

6.2 Surface Areas of Prisms6.2 Surface Areas of Prisms

6.3 Surface Areas of Cylinders6.3 Surface Areas of Cylinders

6.4 Surface Areas of Pyramids6.4 Surface Areas of Pyramids

6.5 Surface Areas of Cones6.5 Surface Areas of Cones

6.6 Surface Areas of Composite Solids6.6 Surface Areas of Composite Solids

“Dear Sir: Why do you sell dog food in tall cans and sell cat food in short cans?”“Neither of these shapes is the optimal use of surface area when compared to volume.”

“I want to paint my dog house. To make sure

I buy the correct amount of paint, I want to

calculate the lateral surface area.”

“Then, because I want to paint the inside and

the outside, I will multiply by 2. Does this

seem right to you?”

6

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Example 1 Find the area.

10 in.

10 in.

3 in. Area = Area of square + Area of triangle

A = s 2 + 1

— 2

bh

= 102 + ( 1 — 2

⋅ 10 ⋅ 3 )

= 100 + 15

= 115 in.2

Find the area.

1.

15 m

8 m

2.

5 cm

4 cm

9 cm

14 cm

Example 2 Find the area. Example 3 Find the area.

7 mmA = πr 2

24 yd

A = πr 2

≈ 3.14(7)2 ≈ 3.14(12)2

= 3.14 ⋅ 49 = 3.14 ⋅ 144

= 153.86 mm2 = 452.16 yd2

Find the area.

3.

5 ft

4.

26 in.

5.

7 cm

What You Learned Before

“Name these shapes.”

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250 Chapter 6 Surface Areas of Solids

Drawing 3-Dimensional Figures6.1

How can you draw three-dimensional fi gures?

Dot paper can help you draw three-dimensional fi gures, or solids. Shading parallel sides the same color helps create a three-dimensional illusion.

Square Dot Paper Isometric Dot Paper

Face-On View Corner View

Work with a partner.

Draw the front, side, and top views of each stack of cubes. Then fi nd the surface area and volume. Each small cube has side lengths of 1 unit.

a. Sample:

top front side

Volume: 3 cubic units

Surface Area: 14 square units

b. c. d.

e. f. g.

ACTIVITY: Finding Surface Areas and Volumes11

COMMON CORE STATE STANDARDS

7.G.3

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Section 6.1 Drawing 3-Dimensional Figures 251


a. Draw all the different solids you can make by joining four cubes. (Two have been drawn.) Cubes must be joined on faces, not on edges only. Translations, refl ections, and rotations do not count as different solids.

b. Do all the solids have the same surface area? Do all the solids have the same volume? Explain your reasoning.

ACTIVITY: Drawing Solids22

Use what you learned about three-dimensional fi gures to complete Exercises 7–9 on page 254.

Same solid reflected

3. IN YOUR OWN WORDS How can you draw three-dimensional fi gures? Draw and shade two prisms that have the same volume but different surface areas.

4. Maurits Escher (1898 –1972) was a popular artist who drew optical illusions.

a. What is the illusion in Escher’s drawing?

b. Why is the cartoon funny? What is the illusion in the cartoon?

©2010 M.C. Escher’s “Ascending and Descending”



Lesson6.1

A three-dimensional fi gure, or solid, has length, width, and depth. A polyhedron is a three-dimensional fi gure whose faces are all polygons.

A cylinder is a solid that has two parallel, identical circular bases.

BasesLateralsurface

A cone is a solid that has one circular base and one vertex.

Vertex

Base

Lateralsurface

A prism is a polyhedron that has two parallel, identical bases. The lateral faces are parallelograms.

Lateral face

Base

Base

Triangular Prism

A pyramid is a polyhedron that has one base. The lateral faces are triangles.

Base

Lateral face

Rectangular Pyramid

The shape of the base tells the name of the prism or the pyramid.

EXAMPLE Drawing a Prism11Draw a rectangular prism.

Step 1 Step 2 Step 3

Draw identical Connect corresponding Change any hiddenrectangular bases. vertices. lines to dashed lines.

Lesson Tutorials

Key Vocabularythree-dimensional fi gure, p. 252polyhedron, p. 252lateral face, p. 252

Prisms Pyramids

Cylinders Cones



EXAMPLE Drawing a Pyramid22Draw a triangular pyramid.

Step 1 Step 2 Step 3

Draw a triangular Connect the vertices of Change any hiddenbase and a point. the triangle to the point. lines to dashed lines.

Draw the solid.

1. Square prism 2. Pentagonal pyramid

EXAMPLE Drawing Views of a Solid33Draw the front, side, and top views of the paper cup.

The front view is The side view is The top view isa triangle. a triangle. a circle.

Draw the front, side, and top views of the solid.

3. 4. 5.

Exercises 10 –15

Exercises 16 –21


Exercises6.1


1. VOCABULARY Compare and contrast prisms and cylinders.

2. VOCABULARY Compare and contrast pyramids and cones.

3. WRITING Give examples of prisms, pyramids, cylinders, and cones in real life.

Identify the shape of the base. Then name the solid.

4. 5. 6.

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

Draw the front, side, and top views of the stack of cubes. Then fi nd the surface area and volume.

7. 8. 9.

Draw the solid.

10. Triangular prism 11. Pentagonal prism 12. Rectangular pyramid

13. Hexagonal pyramid 14. Cone 15. Cylinder

Draw the front, side, and top views of the solid.

16. 17. 18.

19. 20. 21.

11 22

33

Help with Homework



Find the area. (Skills Review Handbook)

29.

4 m

7 m

30. 3 cm

8 cm

31. 6 ft

4 ft

3 ft

32. MULTIPLE CHOICE You borrow $200 and agree to repay $240 at the end of 2 years. What is the simple interest rate per year? (Section 4.4)

○A 5% ○B 10% ○C 15% ○D 20%

22. PYRAMID ARENA The Pyramid of Caius Cestius in Rome is in the shape of a square pyramid. Draw a sketch of the pyramid.

23. RESEARCH Use the Internet to fi nd a picture of the Washington Monument. Describe its shape.

Draw a solid with the following front, side, and top views.

24.

front side top

25.

front side top

26. PROJECT Design and draw a house. Name the different solids that can be used to make a model of the house.

27. REASONING Two of the three views of a solid are shown.

a. What is the greatest number of unit cubes in the solid?

b. What is the least number of unit cubes in the solid? top

sidec. Draw the front views of both solids in parts (a) and (b).

28. Draw two different solids with fi ve faces.

a. Write the number of vertices and edges for each solid.

b. Explain how knowing the numbers of edges and vertices helps you draw a three-dimensional fi gure.



Surface Areas of Prisms6.2

How can you use a net to fi nd the surface

area of a prism?


a. Use the net for the rectangular prism to fi nd its surface area.

5

Base Net

Base

Lateral Face

Lateral Face

Lateral Face

Lateral Face

6

3

b. Copy the net for a rectangular prism. Label each side as h, w, or ℓ. Then use your drawing to write a formula for the surface area of a rectangular prism.

h

w

ACTIVITY: Surface Area of a Right Rectangular Prism11

Lateral Face

Base

Rectangular Prism Triangular Prism

The surface area of a prism is the sum of the areas of all its faces. A two-dimensional representation of a solid is called a net .

Lateral Face

BaseCOMMON CORE STATE STANDARDS

7.G.47.G.6


Section 6.2 Surface Areas of Prisms 257

Work with a partner. Find the surface area of the solid shown by the net. Copy the net, cut it out, and fold it to form a solid. Identify the solid.

4

5

3

4

4

3

3

3

ACTIVITY: Finding Surface Area22

Use what you learned about the surface area of a prism to complete Exercises 6 – 8 on page 260.

3. IN YOUR OWN WORDS How can you use a net to fi nd the surface area of a prism? Draw a net, cut it out, and fold it to form a prism.

4. The greater the surface area of an ice block, the faster it will melt. Which will melt faster, the bigger block or the three smaller blocks? Explain your reasoning.

1 ft

3 ft

1 ft

1 ft

1 ft

1 ft



Lesson6.2

Surface Area of a Rectangular Prism

Words The surface area S of a rectangular prism is the sum of the areas of the bases and the lateral faces.

h

w

base

base

lateral face

lateral face

lateral face

lateral face

www

h

Algebra S = 2ℓw + 2ℓh + 2wh

EXAMPLE Finding the Surface Area of a Rectangular Prism11Find the surface area of the prism.

Draw a net.

S = 2ℓw + 2ℓh + 2wh

= 2(3)(5) + 2(3)(6) + 2(5)(6)

= 30 + 36 + 60

= 126

The surface area is 126 square inches.

Find the surface area of the prism.

1.

3 ft2 ft

4

2.

5 m

8 m

8 m

Area ofbases

Area oflateral faces

6 in.

3 in.

5 in.

5 in.

3 in.

6 in.

3 in.5 in.5 in.

Exercises 9–11

Lesson Tutorials

Key Vocabularysurface area, p. 256net, p. 256

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Surface Area of a Prism

The surface area S of a prism is the sum of the areas of the bases and the lateral faces.

S = areas of bases + areas of lateral faces

EXAMPLE Finding the Surface Area of a Triangular Prism22Find the surface area of the prism.

Draw a net. 4 m3 m

5 m4 m

6 m

3 m

Add the areas of the bases and the lateral faces.

S = areas of bases + areas of lateral faces

= 6 + 6 + 18 + 30 + 24

= 84

The surface area is 84 square meters.


3. 12 m5 m

13 m

3 m

4.

3 cm4 cm

5 cm

4 cm

Areas of lateral faces

Green lateral face: 3 ⋅ 6 = 18

Purple lateral face: 5 ⋅ 6 = 30

Blue lateral face: 4 ⋅ 6 = 24

Area of a base

Red base: 1

— 2

⋅ 3 ⋅ 4 = 6

5 m

6 m

4 m3 m

RememberThe area A of a triangle with base b and height

h is A = 1 —

2 bh.

Exercises 12–14

There are two identical bases. Count the area twice.


Exercises6.2


5 m

5 m

7 m6 m

4 m

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

Draw a net for the prism. Then fi nd the surface area.

6.

5 in.4 in.

3 in. 7.

3 cm

5 cm

4 cm

5 cm

8. 7 m

6 m3 m


9. 1 ft

5 ft10 ft

10. 3 cm

6 cm

9 cm

11. 5 yd

4 yd2 yd

12. 2.2 ft

3 ft1 ft

2 ft

13. 14.

3 mm

4 mm4 mm

5.7 mm

15. GIFT BOX What is the least amount of wrapping paper needed to wrap a gift box that measures 8 inches by 8 inches by 10 inches? Explain.

16. TENT What is the least amount of fabric needed to make the tent?

1. OPEN-ENDED Describe a real-world situation in which you would want to fi nd the surface area of a prism.

Find the indicated area for the rectangular prism.

4 cm

6 cm3 cm

Face C

Face A

Face B

2. Area of Face A

3. Area of Face B

4. Area of Face C

5. Surface area of the prism

5 ft

5 ft

6 ft7 ft 4 ft

11

22

Help with Homework



Find the perimeter. (Skills Review Handbook)

24. 7 8

10

25.

12

12

12

12 26. 1111

914

9

27. MULTIPLE CHOICE The class size increased 25% to 40 students. What was the original class size? (Section 4.2)

○A 10 ○B 30 ○C 32 ○D 50


17. 12 in.

3 in.

5 in.6 in.

5 in.

4 in. 18. 2.5 m

4 m4 m

2 m

19. AQUARIUM A public library has an aquarium in the shape of a rectangular prism. The base is 6 feet by 2.5 feet. The height is 4 feet. How many square feet of glass were used to build the aquarium? (The top of the aquarium is open.)

20. STORAGE BOX The material used to make a storage box costs $1.25 per square foot. The boxes have the same volume. How much does a company save by choosing to make 50 of Box 2 instead of 50 of Box 1?

21. RAMP A quart of stain covers 100 square feet. How many quarts should you buy to stain the wheelchair ramp? (Assume you do not have to stain the bottom of the ramp.)

22. LABEL A label that wraps around a box of golf balls covers 75% of its lateral surface area. What is the value of x?

23. Write a formula for the surface area of a rectangular prism using the height h, the perimeter P of a base, and the area B of a base.

Length Width Height

Box 1 20 in. 6 in. 4 in.

Box 2 15 in. 4 in. 8 in.

25 ft

25 ft

5 ft

25 in.

112

3 in.

2 in.

2 in.x


261A Chapter 6 Surface Areas of Solids

Lesson Tutorials

Radius and Diameter

Words The diameter d of a circle is twice the radius r. The radius r of a circle is one-half the diameter d.

Algebra Diameter: d = 2r Radius: r = d

— 2

EXAMPLE Finding a Radius and a Diameter11

a. The diameter of a circle is b. The radius of a circle is 12 feet. Find the radius. 8 meters. Find the diameter.

12 ft

8 m

r = d

— 2

Radius of a circle d = 2r Diameter of a circle

= 12

— 2

Substitute 12 for d. = 2(8) Substitute 8 for r.

= 6 Divide. = 16 Multiply.

The radius is 6 feet. The diameter is 16 meters.

A circle is the set of all points in a plane that are the same distance from a point called the center.

1. DIAMETER The radius of a dartboard is 9 inches. Find the diameter.

2. RADIUS The diameter of a clock is 1 foot. Find the radius.

centercircle

The radius is the distancefrom the center to anypoint on the circle. The diameter is the

distance across the circlethrough the center.

Circles6.2b

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Circles 261B

Find (a) the circumference and (b) the area of the sticker. Use 3.14 for 𝛑.

a. C = 2π r b. A = πr 2

≈ 2 ⋅ 3.14 ⋅ 3 Substitute. ≈ 3.14 ⋅ (3)2

= 6.28 ⋅ 3 Simplify. = 3.14 ⋅ 9

= 18.84 Simplify. = 28.26

The circumference is The area is aboutabout 18.84 centimeters. 28.26 square centimeters.

EXAMPLE Finding the Circumference and Area of a Circle22

Find the circumference and area of the object. Use 3.14 or 22

— 7

for 𝛑.

3.

70 cm

4.

24 in.

5.

5 in.

6. TIRE The diameter of a bicycle tire is 26 inches.

a. Find the circumference of the tire. Use 3.14 for π.

b. How many rotations does the tire make to travel 95 feet? Explain your reasoning.

Circumference of a Circle

Words The circumference C of a circle is equal to the product of π and the diameter d or the product of π and twice the radius r.

Algebra C = π d or C = 2π r

Area of a Circle

Words The area A of a circle is the product of π and the square of the radius r.

Algebra A = π r 2

The distance around a circle is called the circumference. The ratio

circumference

—— diameter

is the same for every circle and is represented by the Greek

letter π, called pi. The value of π can be approximated as 3.14 or 22

— 7

.Study TipWhen the radius ordiameter is a multiple of 7, it is easier to use

22

— 7 as the estimate of π.

C

d

r

3 cm

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Surface Areas of Cylinders6.3

How can you fi nd the surface area of

a cylinder?

Work with a partner. Use a cardboard cylinder.

● Talk about how you can fi nd the area of the outside of the roll.

● Use a ruler to estimate the area of the outside of the roll.

● Cut the roll and press it out fl at. Then fi nd the area of the fl at-tened cardboard. How close is your estimate to the actual area?

ACTIVITY: Finding Area11

The surface area of a cylinder is the sum of the areas of the bases and the lateral surface.


● Trace the top and bottom of a can on paper. Cut out the two shapes.

● Cut out a long paper rectangle. Make the width the same as the height of the can. Wrap the rectangle around the can. Cut off the excess paper so the edges just meet.

● Make a net for the can. Name the shapes in the net.

● How are the dimensions of the rectangle related to the dimensions of the can?

● Explain how to use the net to fi nd the surface area of the can.

ACTIVITY: Finding Surface Area22

Cut

Base

Lateralsurface

r

h

Base


7.G.47.G.6


Section 6.3 Surface Areas of Cylinders 263

Work with a partner. From memory, estimate the dimensions of the real-life item in inches. Then use the dimensions to estimate the surface area of the item in square inches.

a. b. c.

d.

ACTIVITY: Estimation33

4. IN YOUR OWN WORDS How can you fi nd the surface area of a cylinder? Give an example with your description. Include a drawing of the cylinder.

5. To eight decimal places, π ≈ 3.14159265. Which of the following is closest to π ?

a. 3.14 b. 22

— 7

c. 355

— 113

Use what you learned about the surface area of a cylinder to complete Exercises 5 – 7 on page 266.

“To approximate the irrational number 3.141593, I simply

remember 1, 1, 3, 3, 5, 5.”

“Then I compute the rational number 3.141593.”



Lesson6.3

Area, A = r 2π Circumference, C = 2 rπ

Radius, rThe diagram reviews some important facts for circles.

Surface Area of a Cylinder

Words The surface area S of a cylinder is the sum of the areas of the bases and the lateral surface.

Algebra S = 2π r 2 + 2π rh

Area of lateral surface

Area of bases

EXAMPLE Finding the Surface Area of a Cylinder11Find the surface area of the cylinder. Round your answer to the nearest tenth.

Draw a net.

S = 2π r 2 + 2π rh

= 2π (4)2 + 2π (4)(3)

= 32π + 24π

= 56π ≈ 175.8

The surface area is about 175.8 square millimeters.

1. A cylinder has a radius of 2 meters and a height of 5 meters. Find the surface area of the cylinder. Round your answer to the nearest tenth.

Exercises 8 –10

Remember

π = circumference

—— diameter

Pi can be approximated

as 3.14 or 22

— 7 .

r

h

Base

Lateral surface

r

h

2 r

Base

π

r

4 mm

3 mm

3 mm

4 mm

4 mm

Lesson Tutorials



EXAMPLE Finding Surface Area22How much paper is used for the label on

2 in.

1 in. the can of peas?

Find the lateral surface area of the cylinder.

S = 2π rh

= 2π (1)(2) Substitute.

= 4 π ≈ 12.56 Multiply.

About 12.56 square inches of paper is used for the label.

Do not include the area of the bases in the formula.

EXAMPLE Real-Life Application33You earn $0.01 for recycling the can in Example 2. How much can you expect to earn for recycling the tomato can? Assume that the recycle value is proportional to the surface area.

Find the surface area of each can.

Tomatoes Peas

S = 2π r 2 + 2π rh S = 2π r 2 + 2π rh

= 2π (2)2 + 2π (2)(5.5) = 2π (1)2 + 2π (1)(2)

= 8 π + 22π = 2π + 4 π

= 30π = 6 π

Use a proportion to fi nd the recycle value x of the tomato can.

30 π in.2

— x

= 6π in.2

— $0.01

30π ⋅ 0.01 = x ⋅ 6 π Use Cross Products Property.

5 ⋅ 0.01 = x Divide each side by 6π.

0.05 = x Simplify.

You can expect to earn $0.05 for recycling the tomato can.

2. WHAT IF? In Example 3, the height of the can of peas is doubled.

a. Does the amount of paper used in the label double?

b. Does the recycle value double? Explain.

surface area

recycle value

Exercises 11–13

5.5 in.

2 in.


Exercises6.3

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=


1. CRITICAL THINKING Which part of the formula S = 2π r 2 + 2π r h represents the lateral surface area of a cylinder?

2. CRITICAL THINKING Given the height and the circumference of the base of a cylinder, describe how to fi nd the surface area of the entire cylinder.

Find the indicated area of the cylinder.

3. Area of a base

4. Surface area

Make a net for the cylinder. Then fi nd the surface area of the cylinder. Round your answer to the nearest tenth.

5. 3 ft

2 ft

6.

1 m

4 m 7. 7 ft

5 ft

Find the surface area of the cylinder. Round your answer to the nearest tenth.

8. 5 mm

2 mm

9. 6 ft

7 ft

10.

Find the lateral surface area of the cylinder. Round your answer to the nearest tenth.

11. 10 ft

6 ft

12. 9 in.

4 in.

13.

2 m

14 m

14. TANKER The truck’s tank is a stainless steel cylinder. Find the surface area of the tank.

11

22

6 cm

3 cm

6 cm

12 cm

50 ft

radius 4 ft

Help with Homework


1 in.

3 in.


Evaluate the expression. (Skills Review Handbook)

21. 1

— 2

(26)(9) 22. 1

— 2

(8.24)(3) + 8.24 23. 1

— 2

(18.84)(3) + 28.26

24. MULTIPLE CHOICE A store pays $15 for a basketball. The percent of markup is 30%. What is the selling price? (Section 4.3)

○A $10.50 ○B $19.50 ○C $30 ○D $34.50

15. ERROR ANALYSIS Describe and 6 ft

11 ft

correct the error in fi nding the surface area of the cylinder.

16. OTTOMAN What percent of the surface area of the ottoman is green (not including the bottom)?

17. REASONING You make two cylinders using 8.5-inch by 11-inch pieces of paper. One has a height of 8.5 inches and the other has a height of 11 inches. Without calculating, compare the surface areas of the cylinders.

18. INSTRUMENT A ganza is a percussion instrument used in samba music.

a. Find the surface area of each of the two labeled ganzas.

b. The weight of the smaller ganza is 1.1 pounds. Assume that the surface area is proportional to the weight. What is the weight of the larger ganza?

19. BRIE CHEESE The cut wedge represents one-eighth of the cheese.

a. Find the surface area of the cheese before it is cut.

b. Find the surface area of the remaining cheese after the wedge is removed. Did the surface area increase, decrease, or remain the same?

20. The lateral surface area of a cylinder is 184 square centimeters. The radius is 9 centimeters. What is the surface area of the cylinder? Explain how you found your answer.

3.5 cm5.5 cm

24.5

cm

10 cm

S = 2𝛑 rh ≈ 2𝛑 (6)(11) = 132𝛑 ft2

✗

16 in.

6 in.

8 in.



6 Study Help

Make a four square to help you study these topics.

1. polyhedron 2. prism

3. pyramid 4. cylinder

5. cone 6. drawing a solid

7. surface area

a. of a prism b. of a cylinder

After you complete this chapter, make foursquares for the following topics.

8. surface area

a. of a pyramid b. of a cone c. of a composite solid

You can use a four square to organize information about a topic. Each of the four squares can be a category, such as defi nition, vocabulary, example, non-example, words, algebra, table, numbers, visual, graph, or equation. Here is an example of a four square for a three-dimensional fi gure.

Rectangularpyramid

Cylinder

Triangular prism Triangle Square

Rectangle CircleCone

Examples Non-examples

DefinitionA figure that has length, width,and depth

Vocabulary• also known as a solid• known as a polyhedron if all of the faces are polygons

Three-DimensionalFigure

“My four square shows that my new red skateboard is faster than my old

blue skateboard.”

Graphic Organizer

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Sections 6.1–6.3 Quiz 269

Quiz6.1– 6.3

Draw the front, side, and top views of the solid. (Section 6.1)

1. 2. 3.

Find the surface area of the cylinder. Round your answer to the nearest tenth. (Section 6.3)

4. 10 ft3 ft

5.

6 m

5 m

Find the surface area of the prism. (Section 6.2)

6.

5 cm

10 cm

4 cm3 cm

7.

7 mm2 mm

4 mm

8. MAILING TUBE What is the least amount of material

3 in.

3 ft

needed to make the mailing tube? (Section 6.3)

9. GEOMETRY Consider a prism that has n faces. Write an expression that represents the number of lateral faces. (Section 6.2)

10. TOMATO PASTE How much more paper is used for the label of the large can of tomato paste than for the label of the small can? (Section 6.3)

11. WOODEN CHEST All the faces of the wooden chest will be painted except for the bottom. Find the area to be painted, in square inches. (Section 6.2)

3 in.

4.5 in.

1 in.

1.5 in.

Progress Check

4 ft 1.5 ft

2 ft

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Surface Areas of Pyramids6.4

How can you fi nd the surface area of

a pyramid?

Even though many well-known pyramids have square bases, the base of a pyramid can be any polygon.

Lateral face

Vertex

Slant height

Base

Triangular Base Square Base Hexagonal Base

Work with a partner. Each pyramid has a square base.

● Draw a net for a scale model of one of the pyramids. Describe your scale.

● Cut out the net and fold it to form a pyramid.

● Find the lateral surface area of the real-life pyramid.

a. Cheops Pyramid in Egypt b. Muttart Conservatory in Edmonton

Side = 230 m, Slant height ≈ 186 m Side = 26 m, Slant height ≈ 27 m

c. Louvre Pyramid in Paris d. Pyramid of Caius Cestius in Rome

Side = 35 m, Slant height ≈ 28 m Side = 22 m, Slant height ≈ 29 m

ACTIVITY: Making a Scale Model11


7.G.6


Section 6.4 Surface Areas of Pyramids 271

Work with a partner. There are many different types of gemstone cuts. Here is one called a brilliant cut.

The size and shape of the pavilion can be approximated by an octagonal pyramid.

a. What does octagonal mean?

b. Draw a net for the pyramid.

c. Find the lateral surface area of the pyramid.

ACTIVITY: Estimation22

Work with a partner. The skylight has 12 triangular pieces of glass. Each piece has a base of 1 foot and a slant height of 3 feet.

a. How much glass will you need to make the skylight?

b. Can you cut the 12 glass triangles from a sheet of glass that is 4 feet by 8 feet? If so, draw a diagram showing how this can be done.

ACTIVITY: Building a Skylight33

4. IN YOUR OWN WORDS How can you fi nd the surface area of a pyramid? Draw a diagram with your explanation.

Use what you learned about the surface area of a pyramid to complete Exercises 4 – 6 on page 274.

2 mm

Slant height 4 mm

Crown

Pavilion

Top View Side View Bottom View



Lesson6.4

8 in.5 in.

5 in.

Key Vocabularyregular pyramid, p. 272slant height, p. 272

A regular pyramid is a pyramid whose base is a regular polygon. The lateral faces are triangles. The height of each triangle is the slant height of the pyramid.

Surface Area of a Pyramid

The surface area S of a pyramid is the sum of the areas of the base and the lateral faces.

S = area of base + areas of lateral faces

EXAMPLE Finding the Surface Area of a Square Pyramid11Find the surface area of the regular pyramid.

Draw a net.

Area of base Area of a lateral face

5 ⋅ 5 = 25 1

— 2

⋅ 5 ⋅ 8 = 20

Find the sum of the areas of the base and the lateral faces.


= 25 + 20 + 20 + 20 + 20

= 105

The surface area is 105 square inches.

1. What is the surface area of a square pyramid with a base side length of 9 centimeters and a slant height of 7 centimeters?

i h

There are 4 identical lateral faces. Count the area 4 times.

slant height

lateral faces

slant height

lateral faces

base

5 in.

8 in.

Lesson Tutorials

RememberIn a regular polygon, all of the sides have the same length and all of the angles have the same measure.



10 m8.7 m

14 m

EXAMPLE Finding the Surface Area of a Triangular Pyramid22Find the surface area of the regular pyramid.

Draw a net.


1

— 2

⋅ 10 ⋅ 8.7 = 43.5 1

— 2

⋅ 10 ⋅ 14 = 70

Find the sum of the areas of the base and the lateral faces.


= 43.5 + 70 + 70 + 70

= 253.5

The surface area is 253.5 square meters.

There are 3 identical lateral faces. Count the area 3 times.

EXAMPLE Real-Life Application33A roof is shaped like a square pyramid. One bundle of shingles covers 25 square feet. How many bundles should you buy to cover the roof ?

The base of the roof does not need shingles. So, fi nd the sum of the areas of the lateral faces of the pyramid.

Area of a lateral face

1

— 2

⋅ 18 ⋅ 15 = 135

There are four identical lateral faces. So, the sum of the areas of the lateral faces is

135 + 135 + 135 + 135 = 540.

Because one bundle of shingles covers 25 square feet, it will take 540 ÷ 25 = 21.6 bundles to cover the roof.

So, you should buy 22 bundles of shingles.

2. What is the surface area of the pyramid at the right?

3. WHAT IF? In Example 3, one bundle of shingles covers 32 square feet. How many bundles should you buy to cover the roof ?

Exercises 4–12

8.7 m

10 m

14 m

15 ft

18 ft

5.2 ft6 ft6 ft

6 ft

10 ft


20 mm

16 mmArea of baseis 440.4 mm .2

Exercises6.4

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=


1. VOCABULARY Which of the polygons could be the base for a regular pyramid?

2. VOCABULARY Can a pyramid have rectangles as lateral faces? Explain.

3. CRITICAL THINKING Why is it helpful to know the slant height of a pyramid to fi nd its surface area?

Use the net to fi nd the surface area of the regular pyramid.

4.

4 in.

3 in. 5. 9 mm

10 mm

Area of baseis 43.3 mm2.

6.

6 m

6 m

Area of baseis 61.9 m2.

In Exercises 7–11, fi nd the surface area of the regular pyramid.

7.

6 ft

9 ft 8. 6 cm

4 cm

9.

9 yd7.8 yd

10 yd

10.

15 in.13 in.

10 in. 11.

12. LAMPSHADE The base of the lampshade is a regular hexagon with a side length of 8 inches. Estimate the amount of glass needed to make the lampshade.

13. GEOMETRY The surface area of a square pyramid is 85 square meters. The base length is 5 meters. What is the slant height?

11 22

33

10 in.

Help with Homework


Pyramid height


Find the area and circumference of the circle. Use 3.14 for 𝛑 . (Skills Review Handbook)

19. 12

20. 8

21. 27

22. MULTIPLE CHOICE A youth baseball diamond is similar to a professional baseball diamond. The ratio of the perimeters is 2 : 3. The distance between bases on a youth diamond is 60 feet. What is the distance between bases on a professional diamond? (Section 5.3)

○A 40 ft ○B 90 ft ○C 120 ft ○D 180 ft

14. BMX You are building a bike ramp that is shaped like a square pyramid. You use two 4-foot by 8-foot sheets of plywood. How much plywood do you have left over?

15. UMBRELLA You are making an umbrella that is shaped like a regular octagonal pyramid.

a. Estimate the amount of fabric that is needed to make the umbrella.

b. The fabric comes in rolls that are 72 inches wide. You don’t want to cut the fabric “on the bias”. Find out what this means. Then, draw a diagram of how you can cut the fabric most effi ciently.

c. How much fabric is wasted?

16. PRECISION The height of a pyramid is the distance between the base and the top of the pyramid. Which is greater, the height of a pyramid or the slant height? Explain your reasoning.

17. TETRAHEDRON A tetrahedron is a triangular pyramid whose four faces are identical equilateral triangles. The total lateral surface area is 93 square centimeters. Find the surface area of the tetrahedron.

18. Is the total area of the lateral faces of a pyramid greater than,

less than, or equal to the area of the base? Explain.

4 ft

5 ft

5 ft5 ft

3 ft

ver?

brella that is id.

eded t



Surface Areas of Cones6.5

How can you fi nd the surface area of a cone?

A cone is a solid with one circular base and one vertex.

Slant height,

Vertex

Height, h

Radius, r

ACTIVITY: Finding the Surface Area of a Cone11Work with a partner.

● Draw a circle with a radius of 3 inches. ππ

3 in.

π

π

π π

π

● Mark the circumference of the circle into six equal parts.

● The circumference of the circle is 2(𝛑)(3) = 6𝛑 . So each of the six parts on the circle has a length of 𝛑 . Label each part.

● Cut out one part as shown. Then, make a cone.

π

π

π π

π

a. The base of the cone should be a circle. Explain why the circumference of the base is 5π.

b. Find the radius of the base.

c. What is the area of the original circle?

d. What is the area of the circle with one part missing?

e. Describe the surface area of the cone. Use your description to fi nd the surface area, including the base.


7.G.47.G.6


Section 6.5 Surface Areas of Cones 277


● Cut out another part from the circle in Activity 1 and make a cone.

● Find the radius of the base and the surface area of the cone.

● Record your results in the table.

● Repeat this three times.

● Describe the pattern.

ACTIVITY: Experimenting with Surface Area22

Write a story that uses real-life cones. Include a diagram and label the dimensions. In your story, explain why you would want to know the surface area of the cone. Then, estimate the surface area.

ACTIVITY: Writing a Story33

4. IN YOUR OWN WORDS How can you fi nd the surface area of a cone? Draw a diagram with your explanation.

Use what you learned about the surface area of a cone to complete Exercises 4 –6 on page 280.

Shape

Radius of Base

Slant Height

Surface Area



Lesson6.5

Key Vocabularyslant height, p. 278

The distance from the vertex of a cone to any point on the edge of its base is called the slant height of the cone.

Surface Area of a Cone

Words The surface area S of a cone is the sum of the areas of the base and the lateral surface.

Algebra S = π r 2 + π rℓ

Area of lateral surfaceArea of base

Slant height,

r

Slant height, r

Lateral surface

Base

2 rπ

EXAMPLE Finding the Surface Area of a Cone11

Find the surface area of the cone. Roundyour answer to the nearest tenth.

Draw a net.

S = π r 2 + πrℓ

= π (1)2 + π(1)(3)

= π + 3π

= 4π ≈ 12.6

The surface area is about 12.6 square meters.

Find the surface area of the cone. Round your answer to the nearest tenth.

1.

2 ft 6 ft

2.

4 cm8 cm

Exercises 4 – 9

3 m

1 m

3 m1 m

Lesson Tutorials



EXAMPLE Finding the Slant Height of a Cone22

The surface area of the cone is 100𝛑 square meters. What is the slant height ℓ of the cone?

S = π r 2 + πrℓ Write formula.

100π = π(5)2 + π(5)(ℓ) Substitute.

100π = 25π + 5πℓ Simplify.

75π = 5πℓ Subtract 25π from each side.

15 = ℓ Divide each side by 5π.

The slant height is 15 meters.

EXAMPLE Real-Life Application33You design a party hat. You attach a piece of elastic along a diameter. (a) How long is the elastic? (b) How much paper do you need to make the hat?

a. To fi nd the length of the elastic, fi nd the diameter of the base.

C = πd Write formula.

22 ≈ (3.14)d Substitute.

7.0 ≈ d Solve for d.

The elastic is about 7 inches long.

b. To fi nd how much paper you need, fi nd the lateral surface area.

S = πrℓ

= π(3.5)(5) Substitute.

= 17.5π ≈ 55 Multiply.

You need about 55 square inches of paper to make the hat.

3. WHAT IF? In Example 2, the surface area is 50π square meters. What is the slant height of the cone?

4. WHAT IF? In Example 3, the slant height of the party hat is doubled. Does the amount of paper used double? Explain.

Do not include the area of the base in the formula.

RememberThe diameter d of a circle is two times the radius r.

d = 2r

Exercises 10 –14

5 m

5 in.

C 22 in.

astic? hat?

5 in.

C 22 in


Exercises6.5


1. VOCABULARY Is the base of a cone a polygon? Explain.

2. CRITICAL THINKING In the formula for the surface area of a cone, what does π rℓ represent? What does π r 2 represent?

3. REASONING Write an inequality comparing the slant height ℓ and the radius r of a cone.

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=


4. 6 in.

3 in.

5. 5 m

4 m

6.

9 mm

5 mm

7. 7 ft

10 ft 8. 5 cm

11 cm

9. 8 yd

12 yd

Find the slant height ℓof the cone.

10. S = 33π in.2 11. S = 126π cm2 12. S = 60π ft2

3 in.

12 cm

5 ft

13. NÓN LÁ How much material is needed to make the Nón Lá Vietnamese leaf hat?

14. PAPER CUP A paper cup shaped like a cone has a diameter of 6 centimeters and a slant height of 7.5 centimeters. How much paper is needed to make the cup?

11

22

33 13 in.

20 in.

Help with Homework



Find the area of the shaded region. Use 3.14 for 𝛑 . (Skills Review Handbook)

23. 4 in.

6 in.

15 in.

24.

3 m

5 m

25. 4 ft

8 ft

26. MULTIPLE CHOICE Which best describes a translation? (Section 5.5)

○A a fl ip ○B a slide

○C a turn ○D an enlargement

Find the surface area of the cone with diameter d and slant height ℓ.

15. d = 2 ft 16. d = 12 cm 17. d = 4 yd

ℓ = 18 in. ℓ = 85 mm ℓ = 10 ft

18. ROOF A roof is shaped like a cone with a diameter of 12 feet. One bundle of shingles covers 32 square feet. How many bundles should you buy to cover the roof ?

19. MEGAPHONE Two stickers are placed on opposite sides of the megaphone. Estimate the percent of the surface area of the megaphone covered by the stickers. Round your answer to the nearest percent.

20. REASONING The height of a cone is the distance between the base and the vertex. Which is greater, the height of a cone or the slant height? Explain your reasoning.

21. GEOMETRY The surface area of a cone is also given as S = 1

— 2

Cℓ + B, where

C is the circumference and ℓ is the slant height. What does 1

— 2

Cℓ represent?

22. A cone has a diameter of x millimeters and a slant height of y millimeters. A square pyramid has a base side length of x millimeters and a slant height of y millimeters. Which has the greater surface area? Explain.

13 ft131313113131313131313131313131313113131313131313331313133313131311313131313131333133133333311131313111131111133133111331333131333111111313333313111111133331111113313313113 fffffffffffffffffffffffffffffffffffffffffffffffffffffffttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt13 ft

Cone height

6 in.

6 in. 1.2 ft

2.25 ft



Surface Areas of Composite Solids6.6

How can you fi nd the

surface area of a composite solid?

Work with a partner. You are manufacturing scale models of old houses.

a. Name the four basic solids of this composite fi gure.

b. Determine a strategy for fi nding the surface area of this model. Would you use a scale drawing? Would you use a net?Explain.

ACTIVITY: Finding a Surface Area11

Many castles have cylindrical towers with conical roofs. These are called turrets.

36 in.

24 in.

Top View

18 in.18 in.

16 in.36 in.

60 in.

Front View

in.


7.G.47.G.6


Section 6.6 Surface Areas of Composite Solids 283


● Find the surface area of each fi gure.

● Use a table to organize your results.

● Describe the pattern in the table.

● Use the pattern to fi nd the surface area of the fi gure that has a base of 10 blocks.

ACTIVITY: Finding and Using a Pattern22

Work with a partner. You own a roofi ng company. Each building has the same base area. Which roof would be cheapest? Which would be the most expensive? Explain your reasoning.

ACTIVITY: Finding and Using a Pattern33

4. IN YOUR OWN WORDS How can you fi nd the surface area of a composite solid?

5. Design a building that has a turret and also has a mansard roof. Find the surface area of the roof.

Use what you learned about the surface area of a composite solid to complete Exercises 6 – 8 on page 286.

fl at lean-to gable hip

cross-hipped gambrel mansard cross-gabled



Lesson6.6

20 in. 7 in.10 in.

7 in.

24 in.

A composite solid is a fi gure that is made up of more than one solid.

composite solid

cylinder

cone

Key Vocabularycomposite solid, p. 284

EXAMPLE Standardized Test Practice22

Lesson Tutorials

EXAMPLE Identifying Solids11

Identify the solids that make up Fort Matanzas.

Rectangularprism

Cylinder

Approximately arectangular prism

You painted the steps to an apartment green. What is the surface area that you painted?

○A 210 in.2 ○B 408 in.2 ○C 648 in.2 ○D 1056 in.2

Find the area of each green face.

Green area on top step:A = 20(24) + 7(24) = 648 in.2

Green area on bottom step:A = 10(24) + 7(24) = 408 in.2

20 in.7 in.

10 in.

7 in.

24 in.

24 in.

You painted 648 + 408 = 1056 square inches.

The correct answer is ○D .

1. WHAT IF? In Example 2, you also painted the sides of the steps green. What is the surface area that you painted?Exercises 3–5



EXAMPLE Finding the Surface Area of a Composite Solid33Find the surface area of the composite solid.

The solid is made up of a square prism and a square pyramid. Use the surface area formulas for a prism and a pyramid, but do not include the areas of the sides that overlap.

8 m

Do not include the top base ofthe prism in the surface area.

6 m6 m

Do not include the base of thepyramid in the surface area.

4 m

6 m

6 m

Square prism

S = ℓw + 2ℓh + 2wh Write formula.

= 6(6) + 2(6)(4) + 2(6)(4) Substitute.

= 36 + 48 + 48 Multiply.

= 132 Add.

Square pyramid

S = areas of lateral faces Write formula.

= 4 ( 1 — 2

⋅ 6 ⋅ 8 ) Substitute.

= 96 Multiply.

Find the sum of the surface areas: 132 + 96 = 228.

The surface area is 228 square meters.

Identify the solids that make up the composite solid. Then fi nd the surface area. Round your answer to the nearest tenth.

2.

5 yd

3 yd5 yd

3. 4 cm

5 cm

3 cm5 cm

3 cm

Exercises 6–11

4 m

6 m

6 m

8 m


6 cm

5 cm

3 cm

2 cm

2 cm2.5 cm

2.5 cm

Exercises6.6


1. OPEN-ENDED Draw a composite solid formed by a triangular prism and a cone.

2. REASONING Explain how to fi nd the surface area of

7 in.

10 in.

4 in.

the composite solid.

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

Identify the solids that form the composite solid.

3. 4. 5.

Identify the solids that form the composite solid. Then fi nd the surface area. Round your answer to the nearest tenth.

6. 4 ft

6 ft

6 ft

3 ft

7. 8 m

10 m

8 m

4 m

8.

4 in.

5 in.5 in.

5 in.

4 in.

9. 10.

8 in.

10 in.

6.9 in.

7 in.

8 in.

8 in. 11.

12. OPEN-ENDED The solid is made using eight cubes with side lengths of 1 centimeter.

a. Draw a new solid using eight cubes that has a surface area less than that of the original solid.

b. Draw a new solid using eight cubes that has a surface area greater than that of the original solid.

11

33

8 ft2 ft

4 ft

12 ft5 ft

Help with Homework

22



Find the area. (Skills Review Handbook)

19. 2 ft

5 ft

20. 4 cm

8 cm

21. 7 in.

5 in.

12 in.

22. MULTIPLE CHOICE A cliff swallow nest is 86 meters above a canyon fl oor. The elevation of the nest is −56 meters. What is the elevation of the canyon fl oor? (Section 2.4)

○A −142 ○B −30 ○C 30 ○D 142

13. BATTERIES What is the percent increase in the surface area of the AAA battery to the AA battery? Round your answer to the nearest tenth of a percent.

14. BARBELL The diameter of the handle of a barbell is 1 inch. The hexagonal weights are identical. What is the surface area of the barbell?

REASONING Find the surface area of the solid. Round your answer to the nearest tenth.

15.

10 in.

2 in.4 in. 16.

8 ft4 ft

5 ft

4 ft1 ft 17.

10 mm

24 mm25 mm

7 mm

18. The cube is made with 27 identical cubes. All cubes that cannot be seen are orange. Is the surface area of the solid formed without the purple cubes greater than, less than, or equal to the surface area of the solid formed without the green cubes? Explain your reasoning.

43 mm

10 mm

AAA battery AA battery

49 mm

14 mm

49 mm

1 mm

5.5 mm

0.8 mm

3.8 mm

3.5 in.

1.75 in.

1.5 in.

2 in.

5 in.


Quiz6.4 – 6.6


Identify the solids that form the composite solid. (Section 6.6)

1. 2. 3.

Find the surface area of the regular pyramid. (Section 6.4)

4.

5 m

Area ofbase is

65.0 m2.

12 m

5.

2 cm

6 cm

Find the surface area of the cone. Round your answer to the nearest tenth. (Section 6.5)

6.

3 m 8 m

7. 7 mm

6 mm

Find the surface area of the composite solid. Round your answer to the nearest tenth. (Section 6.6)

8. 3 m

2 m

1 m 9.

5 mm

3 mm

3 mm 2 mm

4 mm

10. TRAFFIC CONE A square refl ective sticker is placed on a traffi c cone to make it more visible at night. Estimate the percent of the surface area of the traffi c cone covered by the sticker to the nearest percent. (Section 6.5)

11. GEOMETRY The surface area of a cone is 150π square inches. The radius of the base is 10 inches. What is the slant height? (Section 6.5)

12. TOOLBOX Find the surface area of the toolbox. (Section 6.6)

25 cm

99 cm

12 cm

12 cm

Progress Check

9.6 in.

8 in. 18 in.

5 in.

1.4 in.

MSCC7PE_0600_ec.indd 288MSCC7PE_0600_ec.indd 288 12/13/11 7:22:25 AM12/13/11 7:22:25 AM

Chapter Review 289

Chapter Review6Review Key Vocabulary

Review Examples and Exercises

Draw the solid.

1. Square pyramid 2. Hexagonal prism 3. Cylinder

6.16.1 Drawing 3-Dimensional Figures (pp. 250–255)

Draw a triangular prism.

Draw identical Connect corresponding Change any hidden triangular bases. vertices. lines to dashed lines.

6.26.2 Surface Areas of Prisms (pp. 256 –261)


Draw a net.

S = 2ℓw + 2ℓh + 2wh

= 2(6)(4) + 2(6)(5) + 2(4)(5)

= 48 + 60 + 40

= 148

The surface area is 148 square feet.

5 ft

6 ft4 ft

Vocabulary Help


4. 4 in.

7 in. 2 in.

5. 17 cm

7 cm8 cm

15 cm

6. 4 m3 m

5 m

8 m

6 ft

4 ft 4 ft

4 ft

6 ft

5 ft

three-dimensional fi gure, p. 252polyhedron, p. 252lateral face, p. 252

surface area, p. 256net, p. 256regular pyramid, p. 272

slant height, pp. 272, 278composite solid, p. 284



6.36.3 Surface Areas of Cylinders (pp. 262 –267)


Draw a net.

S = 2π r 2 + 2π r h

= 2π (8)2 + 2π (8)(9)

= 128π + 144π

= 272π ≈ 854.1



7.

2 in.7 in. 8.

1 cm

3 cm 9. 4 m

5 m

8 mm

9 mm

8 mm

9 mm

8 mm

Find the surface area of the regular pyramid.

10. 3 in.

2 in.

11.

8 m 6.9 m

10 m

12. 9 cm

7 cmArea of baseis 84.3 cm2.

6.46.4 Surface Areas of Pyramids (pp. 270–275)

Find the surface area of the regular pyramid.

Draw a net.


1

— 2

⋅ 6 ⋅ 5.2 = 15.6 1

— 2

⋅ 6 ⋅ 10 = 30

Find the sum of the areas of the base and all 3 lateral faces.

S = 15.6 + 30 + 30 + 30 = 105.6

The surface area is 105.6 square yards.

10 yd

5.2 yd

6 yd

6 yd 5.2 yd

10 yd


Chapter Review 291

6.56.5 Surface Areas of Cones (pp. 276 –281)


Draw a net.

S = π r 2 + π rℓ

= π (3)2 + π (3)(5)

= 9π + 15π

= 24π ≈ 75.4



13.

1 in.4 in.

14. 4 cm

3 cm

15. 4 m8 m

5 mm3 mm

6.66.6 Surface Areas of Composite Solids (pp. 282 –287)

Find the surface area of the composite solid. Round your answer to the nearest tenth.

The solid is made of a cone and a cylinder. Use the surface area formulas. Do not include the areas of the bases that overlap.

Cone Cylinder

S = π rℓ S = π r 2 + 2π r h

= π (6)(10) = π (6)2 + 2π (6)(8)

= 60π ≈ 188.4 = 36π + 96π

= 132π ≈ 414.5

The surface area is about 188.4 + 414.5 = 602.9 square inches.

Find the surface area of the composite solid. Round your answer to the nearest tenth.

16.

6 ft

6 ft

6 ft

4 ft 17.

6 yd

10 yd

1 yd

2 yd8 yd

18.

10 m

5 m3 m

3 mm

5 mm

10 in.

8 in.

6 in.


6 in.

11 in.

8 in. 3 in.

10 in.

14 ft

15.2 ft

19.5 ft6 ft

Draw the solid.

1. Square prism 2. Pentagonal pyramid 3. Cone

Find the surface area of the prism or regular pyramid.

4. 3 ft

5 ft2 ft

5.

1 in.

2 in. 6.

11 m9.5 m

15 m

Find the surface area of the cylinder or cone. Round your answer to the nearest tenth.

7.

2 m

8 m 8.

7 in.

10 in.

9. Draw the front, side, and top views of the solid in Exercise 8.

Identify the solids that form the composite solid. Then fi nd the surface area. Round your answer to the nearest tenth.

10. 9 ft

1 ft7 ft

11. 13 cm

6 cm

1 cm12 cm

5 cm

12. CORN MEAL How much paper is used for the labelof the corn meal container?

13. CRACKER BOX Find the surface area of the cracker box.

14. COSTUME The cone-shaped hat will be part of a costume for a school play. What is the least amount of material needed to make this hat?

15. SKATEBOARD RAMP A quart of paint covers 80 square feet. How many quarts should you buy to paint the ramp with two coats? (Assume you will not paint the bottom of the ramp.)


Chapter Test6

5 in.

4 in.

Test Practice


1. In the fi gure below, △PQR ∼ △STU .

T 18 cm

20 cm16 cm

12 cm

x

U

S

P

Q

R

What is the value of x ? (7.G.1)

A. 9.6 cm C. 13.5 cm

B. 10 2

— 3

cm D. 15 cm

2. The rectangle below is divided into six regions.

2

4

5 3

What is the area of the part of the fi gure that is shaded? (7.NS.3)

F. 23 units2 H. 25 units2

G. 24 units2 I. 28 units2

3. A right rectangular prism and its dimensions are shown below.

10 in.

5 in.

3 in.

What is the total surface area, in square inches, of the right rectangular prism? (7.G.6)

Standardized Test Practice 293

Standardized Test Practice6Test-Taking StrategyAnswer Easy Questions First

“Scan the test and answer the easy questions first. You know area is

measured in square units.”

MSCC7PE_0600_stp.indd 293MSCC7PE_0600_stp.indd 293 12/13/11 7:23:48 AM12/13/11 7:23:48 AM


4. You rode your bicycle 0.8 mile in 2 minutes. You want to know how many miles you could ride in 1 hour, if you ride at the same rate. Which proportion could you use to get your answer? (7.RP.2d)

A. 0.8

— 2

= 60

— x

C. 0.8

— 2

= 30

— x

B. 0.8

— 2

= x

— 60

D. 0.8

— 2

= x

— 30

5. A right square pyramid is shown below.

The square base and one of the triangular faces of the right square pyramid are shown below with their dimensions.

3 in. 3 in.

5 in.

Square Base A Triangular Face

What is the total surface area of the right square pyramid? (7.G.6)

F. 16.5 in.2 H. 39 in.2

G. 31.5 in.2 I. 69 in.2

6. A right circular cylinder with a radius of 3 centimeters and a height of 7 centimeters will be carved out of wood. (7.G.3, 7.G.6)

Part A Draw and label a right circular cylinder with a radius of 3 centimeters and a height of 7 centimeters.

The two bases of the right circular cylinder will be painted blue. The rest of the cylinder will be painted red.

Part B What is the surface area, in square centimeters, that will be painted blue? Show your work and explain your reasoning. (Use 3.14 for π.)

Part C What is the surface area, in square centimeters, that will be painted red? Show your work and explain your reasoning. (Use 3.14 for π.)


Standardized Test Practice 295

7. Anna was simplifying the expression in the box below.

− 3

— 8

⋅ [ 2 — 5

÷ (−4) ] = − 3

— 8

⋅ [ 2 — 5

⋅ ( − 1 —

4 ) ]

= − 3

— 8

⋅ ( − 1 —

10 )

= − 3 —

80

What should Anna do to correct the error that she made? (7.NS.3)

A. Make the product inside the brackets positive.

B. Multiply by −10 instead of − 1

— 10

.

C. Make the fi nal product positive.

D. Multiply by 4 instead of − 1

— 4

.

8. Which equation has the greatest solution? (7.EE.4a)

F. −3x + 9 = −15 H. x

— 2

− 13 = −7

G. 12 = 2x + 28 I. 6 = x

— 3

+ 10

9. A cube has a total surface area of 600 square inches. What is the length, in inches, of each edge of the cube? (7.G.6)

10. A line contains the two points plotted in the coordinate plane below.

y3

2

1

−4

−5

−2

−3

x3 421O−2−3−4

(0, −5)

(2, 1)

Another point on this line can be represented by the ordered pair (−1, y ). What is the value of y ? (7.NS.1b)

A. −11 C. −6

B. −8 D. −2


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Surface Areas of Solids - Mrs. Middleton Algebra & Geometry

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