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In this pdf, we will be learning about the surface area of cylinders. The surface area S of a cylinder is the sum of the areas of the bases and the lateral surface.

1.
English Spanish

6.3 Surface Areas of Cylinders

How can you find the surface area of

S

STATE a cylinder?

STANDARDS

MA.7.G.2.1

1 ACTIVITY: Finding Area

Work with a partner. Use a

cardboard cylinder.

● Talk about how you can find the

area of the outside of the roll.

● Use a ruler to estimate the area

of the outside of the roll. Cut

● Cut the roll and press it out flat.

Then find the area of the flat-

tened cardboard. How close is

your estimate to the actual area?

Base

r

Lateral h The surface area of a cylinder is the sum of

surface the areas of the bases and the lateral surface.

Base

2 ACTIVITY: Finding Surface Area

Work with a partner.

● 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 find the surface area of

the can.

262 Chapter 6 Surface Areas of Solids

6.3 Surface Areas of Cylinders

How can you find the surface area of

S

STATE a cylinder?

STANDARDS

MA.7.G.2.1

1 ACTIVITY: Finding Area

Work with a partner. Use a

cardboard cylinder.

● Talk about how you can find the

area of the outside of the roll.

● Use a ruler to estimate the area

of the outside of the roll. Cut

● Cut the roll and press it out flat.

Then find the area of the flat-

tened cardboard. How close is

your estimate to the actual area?

Base

r

Lateral h The surface area of a cylinder is the sum of

surface the areas of the bases and the lateral surface.

Base

2 ACTIVITY: Finding Surface Area

Work with a partner.

● 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 find the surface area of

the can.

262 Chapter 6 Surface Areas of Solids

2.
English Spanish

3 ACTIVITY: Estimation

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.

4. IN YOUR OWN WORDS How can you find 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 ?

22 355

a. 3.14 b. — c. —

7 113

“To approximate the irrational “Then I compute the rational

number Ƽ 3.141593, I simply number Ƽ 3.141593.”

remember 1, 1, 3, 3, 5, 5.”

Use what you learned about the surface area of a cylinder to

complete Exercises 5 – 7 on page 266.

Section 6.3 Surface Areas of Cylinders 263

3 ACTIVITY: Estimation

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.

4. IN YOUR OWN WORDS How can you find 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 ?

22 355

a. 3.14 b. — c. —

7 113

“To approximate the irrational “Then I compute the rational

number Ƽ 3.141593, I simply number Ƽ 3.141593.”

remember 1, 1, 3, 3, 5, 5.”

Use what you learned about the surface area of a cylinder to

complete Exercises 5 – 7 on page 266.

Section 6.3 Surface Areas of Cylinders 263

3.
English Spanish

6.3 Lesson

Lesson Tutorials

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

The diagram reviews some

important facts for circles. Radius, r

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. Base

Remember r r

2π r

circumference

π = —— h

diameter

Pi can be approximated Lateral surface h

22

as 3.14 or —.

7 Algebra S = 2π r 2 + 2π rh

r

Area of Area of Base

bases lateral surface

EXAMPLE 1 Finding the Surface Area of a Cylinder

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

4 mm

nearest tenth.

3 mm Draw a net.

4 mm

S = 2π r 2 + 2π rh

= 2π (4)2 + 2π (4)(3)

= 32π + 24π 3 mm

= 56π ≈ 175.8

The surface area is about 4 mm

175.8 square millimeters.

1. A cylinder has a radius of 2 meters and a height of 5 meters.

Exercises 8 –10 Find the surface area of the cylinder. Round your answer to

the nearest tenth.

264 Chapter 6 Surface Areas of Solids

6.3 Lesson

Lesson Tutorials

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

The diagram reviews some

important facts for circles. Radius, r

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. Base

Remember r r

2π r

circumference

π = —— h

diameter

Pi can be approximated Lateral surface h

22

as 3.14 or —.

7 Algebra S = 2π r 2 + 2π rh

r

Area of Area of Base

bases lateral surface

EXAMPLE 1 Finding the Surface Area of a Cylinder

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

4 mm

nearest tenth.

3 mm Draw a net.

4 mm

S = 2π r 2 + 2π rh

= 2π (4)2 + 2π (4)(3)

= 32π + 24π 3 mm

= 56π ≈ 175.8

The surface area is about 4 mm

175.8 square millimeters.

1. A cylinder has a radius of 2 meters and a height of 5 meters.

Exercises 8 –10 Find the surface area of the cylinder. Round your answer to

the nearest tenth.

264 Chapter 6 Surface Areas of Solids

4.
English Spanish

EXAMPLE 2 Finding Surface Area

How much paper is used for the label on 1 in.

the can of peas?

Find the lateral surface area of the cylinder.

2 in.

Do not include the area of

S = 2π rh the bases in the formula.

= 2π (1)(2) Substitute.

= 4 π ≈ 12.56 Multiply.

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

EXAMPLE 3 Real-Life Application

You earn $0.01 for recycling the can in Example 2. How much can you

2 in. 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 + 2π rh

2

S = 2π r 2 + 2π rh

5.5 in.

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

= 8 π + 22π = 2π + 4 π

= 30π = 6π

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

surface area

30 π in.2 6π in.2

—=—

x $0.01 recycle value

⋅ ⋅

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

Exercises 11–13 is doubled.

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

b. Does the recycle value double? Explain.

Section 6.3 Surface Areas of Cylinders 265

EXAMPLE 2 Finding Surface Area

How much paper is used for the label on 1 in.

the can of peas?

Find the lateral surface area of the cylinder.

2 in.

Do not include the area of

S = 2π rh the bases in the formula.

= 2π (1)(2) Substitute.

= 4 π ≈ 12.56 Multiply.

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

EXAMPLE 3 Real-Life Application

You earn $0.01 for recycling the can in Example 2. How much can you

2 in. 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 + 2π rh

2

S = 2π r 2 + 2π rh

5.5 in.

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

= 8 π + 22π = 2π + 4 π

= 30π = 6π

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

surface area

30 π in.2 6π in.2

—=—

x $0.01 recycle value

⋅ ⋅

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

Exercises 11–13 is doubled.

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

b. Does the recycle value double? Explain.

Section 6.3 Surface Areas of Cylinders 265

5.
English Spanish

6.3 Exercises

Help with Homework

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 find the surface area of the entire cylinder.

Find the indicated area of the cylinder. 6 cm

3. Area of a base 3 cm

4. Surface area

6)=3

9+(- 3)=

3+(- 9)=

4+(- =

1)

9+(-

Make a net for the cylinder. Then find the surface area of the cylinder. Round your

answer to the nearest tenth.

5. 6. 4m 7.

3 ft 7 ft

2 ft 1m 5 ft

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

1 8. 5 mm 9. 6 ft 10.

12 cm

2 mm 6 cm

7 ft

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

2 11. 12. 9 in. 13. 14 m

10 ft

2m

6 ft 4 in.

50 ft

14. TANKER The truck’s tank

is a stainless steel cylinder.

Find the surface area of

the tank.

radius â 4 ft

266 Chapter 6 Surface Areas of Solids

6.3 Exercises

Help with Homework

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 find the surface area of the entire cylinder.

Find the indicated area of the cylinder. 6 cm

3. Area of a base 3 cm

4. Surface area

6)=3

9+(- 3)=

3+(- 9)=

4+(- =

1)

9+(-

Make a net for the cylinder. Then find the surface area of the cylinder. Round your

answer to the nearest tenth.

5. 6. 4m 7.

3 ft 7 ft

2 ft 1m 5 ft

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

1 8. 5 mm 9. 6 ft 10.

12 cm

2 mm 6 cm

7 ft

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

2 11. 12. 9 in. 13. 14 m

10 ft

2m

6 ft 4 in.

50 ft

14. TANKER The truck’s tank

is a stainless steel cylinder.

Find the surface area of

the tank.

radius â 4 ft

266 Chapter 6 Surface Areas of Solids

6.
English Spanish

✗

15. ERROR ANALYSIS Describe and 6 ft

correct the error in finding the S = 2𝛑 rh

surface area of the cylinder. ≈ 2𝛑 (6)(11)

11 ft

= 132𝛑 ft2

16 in.

16. OTTOMAN What percent of the surface area of the

ottoman is green (not including the bottom)?

6 in.

8 in. 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.

cm cm a. Find the surface area of the

5

10 2 4.

cheese before it is cut.

3 in.

3.5 cm b. Find the surface area of

5.5 cm the remaining cheese after

the wedge is removed. Did

the surface area increase,

decrease, or remain the same?

1 in.

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.

Evaluate the expression. SKILLS REVIEW HANDBOOK

1 1 1

21. —(26)(9) 22. —(8.24)(3) + 8.24 23. —(18.84)(3) + 28.26

2 2 2

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

Section 6.3 Surface Areas of Cylinders 267

✗

15. ERROR ANALYSIS Describe and 6 ft

correct the error in finding the S = 2𝛑 rh

surface area of the cylinder. ≈ 2𝛑 (6)(11)

11 ft

= 132𝛑 ft2

16 in.

16. OTTOMAN What percent of the surface area of the

ottoman is green (not including the bottom)?

6 in.

8 in. 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.

cm cm a. Find the surface area of the

5

10 2 4.

cheese before it is cut.

3 in.

3.5 cm b. Find the surface area of

5.5 cm the remaining cheese after

the wedge is removed. Did

the surface area increase,

decrease, or remain the same?

1 in.

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.

Evaluate the expression. SKILLS REVIEW HANDBOOK

1 1 1

21. —(26)(9) 22. —(8.24)(3) + 8.24 23. —(18.84)(3) + 28.26

2 2 2

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

Section 6.3 Surface Areas of Cylinders 267