?

Oatmeal

r r

ℓ = 2πr

h h

C = 2πr

Oatmeal

ESSENTIAL QUESTION

EXPLORE ACTIVITY

How do you find the surface area of a cylinder?

Modeling the Surface Area of a CylinderJust as you did with a prism, you can use a net to help

you find a formula for the surface area of a cylinder.

The lateral area of a cylinder is the area of the curved

surface that connects the two bases. The net shows

that the lateral surface is a rectangle.

Use the diagram of the oatmeal container and its net.

Imagine unrolling the container’s lateral surface to form a rectangle.

What dimension of the cylinder matches the rectangle’s length?

What dimension of the cylinder matches the rectangle’s height?

Express the lateral area of a cylinder in words and as a formula using

the given variables.

Lateral area = Area of one = Length · Height

L = (C = circumference, h = height)

L = (r = radius, h = height)

Express the total surface area of a cylinder in words and as a formula

using the given variables.

Total surface area = Area of two + Area of one

S = ( B = base area, C = circumference, h = height)

S = (r = radius, h = height)

Reflect1. Communicate Mathematical Ideas How is the process for finding

lateral and total surface area of a cylinder like the process for a prism?

A

B

C

L E S SON

10.2Surface Area of Cylinders

8.7.B

Expressions, equations, and relationships—8.7.B … make connections to the formulas for lateral and total surface area and determine solutions for problems involving … cylinders.

The formula for the area of a circle is A = π r 2 .

The formula for the circumference of a circle is C = 2πr.

275Lesson 10.2

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Finding the Surface Area of a CylinderThe total surface area of a cylinder is the area of the bases plus the lateral area.

Find the lateral and total surface area of the cylinder

to the nearest tenth. Use 3.14 for π.

Find the lateral area.

L = 2πrh

 ≈ 2(3.14)(4)(7)

 ≈ 175.84

The lateral area is about 175.8 square feet.

Find the total surface area.

S = 2π r 2 + L

 ≈ 2(3.14)( 4 2 ) + 175.84

 ≈ 100.48 + 175.84

 ≈ 276.32

The total surface area of the cylinder is about 276.3 square feet.

EXAMPLE 1

STEP 1

STEP 2

Find the lateral and total surface area of each cylinder. Round your

answers to the nearest tenth. Use 3.14 for π.

2. L:

S:

3. L:

S:

YOUR TURN

8.7.B

RadiusBase

Height

Lateral and Total Surface Area of a Cylinder

The lateral area L of a cylinder with

height h and radius r is the circumference

of the base times the height.

L = Ch or L = 2πrh

The total surface area S of a cylinder

with height h and radius r is twice the

area of a base B plus the lateral area L.

S = 2B + L or S = 2π r 2 + 2πrh

Use the formula for the lateral area of a cylinder.

Use the formula for the totalsurface area of a cylinder.

Add.

Substitute.

Simplify.

Simplify.

Substitute.

Unit 3276

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Finding the Surface Area of a Cylinder in a Real-World SituationYou can find many examples of cylinders in your kitchen.

The can shown is 11 inches high and has a

diameter of 9 inches. How many square inches

of paper are needed for the label? How many square

inches of metal are needed to make the entire can?

Round your answers to the nearest whole number.

Use 3.14 for π.

Find the radius of the base.

d = 9 in., so r = 4.5 in.

To find the area of the label, find the lateral area.

L = 2πrh

   ≈ 2(3.14)(4.5)(11)

   ≈ 310.86

It takes about 311 square inches of paper to make the label.

To find the amount of metal, find the total surface area.

S = 2π r 2 + 2πrh

 ≈ 2(3.14)( 4.5 2 ) + 2(3.14)(4.5)(11)

 ≈ 127.17 + 310.86

 ≈ 438.03

It takes about 438 square inches of metal to make the can.

EXAMPLEXAMPLE 2

STEP 1

STEP 2

STEP 3

4. How many square inches of cardboard are needed for

the lateral area of the raisin container shown? What

is the total surface area of the container? Round your

answers to the nearest tenth. Use 3.14 for π.

YOUR TURN

8.7.B

The radius is half

Use the formula for the lateral area of a cylinder.Substitute.

Multiply.

Use the formula for the total surface area of a cylinder.Substitute.

Simplify.

Add.

of the diameter.

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Guided Practice

1. Draw a net of the cylinder. Label the radius and the height. Then

find the lateral area and total surface area. Round your

answers to the nearest tenth. Use 3.14 for π. (Explore Activity and Example 1)

2. A can of tuna fish has a height of 1 inch and a diameter of 3 inches.

How many square inches of paper are needed for the label? How

many square inches of metal are needed to make the can including

the top and bottom? Round your answers to the nearest whole

number. Use 3.14 for π. (Example 2)

It takes about square inches of paper to make the label.

It takes about square inches of metal to make the can.

3. How do you find the total surface area of a cylinder?

ESSENTIAL QUESTION CHECK-IN??

Find the circumference of the circle. Use 3.14 for π.

C = 2πr ≈ 2 · 3.14 · ≈ inches

The lateral area is the circumference times the

height of the can.

L ≈ 31.4 · ≈ in2

Find the area of the two bases. Use 3.14 for π.

2B = 2π r 2 ≈ 2(3.14) ( ) ≈ in2

Add the area of the bases and the lateral area.

S ≈ + ≈ in2

Unit 3278

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Independent Practice10.2

Find the lateral and total surface area of each cylinder. Round your answers

to the nearest tenth. Use 3.14 for π.

4. 5. 6.

7. A container of bread

crumbs has a radius of

4 inches and a height of

8 inches.

8. A rain barrel has a

diameter of 18 inches

and a height of 3 feet.

9. A drum has a diameter of

14 inches and a height of

5.5 inches.

10. Multistep A pipe is 25 inches long and has a diameter of

5 inches. What is the lateral area of the pipe to the nearest

tenth? Use 3.14 for π.

11. Multistep A size D battery has a diameter of 32 millimeters

and a length of 5.6 centimeters. What is the lateral area of the

battery to the nearest square centimeter? Use 3.14 for π.

12. What If? Carol is designing an oatmeal container. Her first

design is a rectangular prism with a height of 12 inches, a

width of 8 inches, and a depth of 3 inches.

a. What is the total surface area of the container to the

nearest square inch? Use 3.14 for π.

b. Carol wants to redesign the package as a cylinder with

the same total surface area as the prism in part a. If the

radius of the cylinder is 2 inches, what is the height of

the cylinder? Round your answer to the nearest inch.

Use 3.14 for π.

13. Vocabulary How do the lateral and total surface area of a cylinder differ?

8.7.B

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Work Area

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7 cm

14. Multiple Representations The formula for the total surface area of a

cylinder is S = 2π r 2 + 2πrh. Explain how you could use the Distributive

Property to write the formula another way.

15. Persevere in Problem Solving The

treasure chest shown is a composite figure.

What is the surface area of the treasure

chest to the nearest square foot? Explain

how you found your answer.

16. Analyze Relationships A square prism has the same

height as the cylinder shown. The perimeter of the prism’s

base equals the circumference of the cylinder’s base.

a. Find the side length of the prism’s base to the

nearest centimeter.

b. Which figure has the greater lateral area? Which has the greater total

surface area? Which has the greater volume? Explain.

17. Communicate Mathematical Ideas A cylinder has a circumference of

16π inches and its height is half the radius of the cylinder. What is the

total surface area of the cylinder? Give your answer in terms of π. Explain

how you found your answer.

FOCUS ON HIGHER ORDER THINKING

Unit 3280

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