10-4 Surface Area of Prisms and Cylinders10-4

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Surface Areas of Prisms and Cylinders

10-4 Surface Area of Prisms and Cylinders

Warm UpFind the volume of each figure to the nearest tenth. Use 3.14 for .

1. rectangular pyramid 7 ft by 8 ft by 10 ft tall

186.7 ft3

12.6 ft3

162 ft3

2. cone with radius 2 ft and height 3 ft

3. triangular pyramid with base area of 54 ft2

and height 9 ft

10-4 Surface Area of Prisms and Cylinders

Problem of the Day

The volume of a 10-meter-tall square pyramid is 120 m3. What is the length of each side of the base?

6 m

10-4 Surface Area of Prisms and Cylinders

Learn to find the surface area of prisms and cylinders.

10-4 Surface Area of Prisms and Cylinders

Vocabulary

netsurface area

10-4 Surface Area of Prisms and Cylinders

If you remove the surface from a three-dimensional figure and lay it out flat, the pattern you make is called a net.

Nets allow you to see all the surfaces of a solid at one time. You can use nets to help you find the surface area of a three-dimensional figure. Surface area is the sum of the areas of all of the surfaces of a figure.

10-4 Surface Area of Prisms and Cylinders

You can use nets to write formulas for the surface area of prisms. The surface area S is the sum of the areas of the faces of the prism. For the rectangular prism shown,

S = lw + lh + wh + lw + lh + wh= 2lw + 2lh + 2wh

10-4 Surface Area of Prisms and Cylinders

SURFACE AREA OF A PRISM

The surface area of a rectangular prism is the sum of the areas of each face.

S = 2lw + 2lh + 2wh

10-4 Surface Area of Prisms and Cylinders

Find the surface area of the prism formed by the net.

Additional Example 1: Finding the Surface Area of a Prism

= 2lw + 2lh + 2wh S

= (2 · 15 · 9) + (2 · 15 · 7) + (2 · 9 · 7) S Substitute.

S = 270 + 210 + 126

S = 606

The surface area of the prism is 606 in2.

Multiply.

Add.

10-4 Surface Area of Prisms and Cylinders

Find the surface area of the prism formed by the net.

= 2lw + 2lh + 2wh S

= (2 · 4 · 6) + (2 · 4 · 3) + (2 · 6 · 3) S Substitute.

S = 48 + 24 + 36

S = 108

The surface area of the prism is 108 in2.

Multiply.

Add.

Check It Out: Example 1

4 in.

6 in.

3 in. 3 in.4 in.

10-4 Surface Area of Prisms and Cylinders

If you could remove the lateral surface from a cylinder, like peeling a label from a can, you would see that it has the shape of a rectangle when flattened out.

You can draw a net for a cylinder by drawing the circular bases (like the ends of a can) and the rectangular lateral surface as shown below. The length of the rectangle is the circumference, 2r, of the cylinder. So the area of the lateral surface is 2rh. The area of each base is r2.

rCircumference of cylinder (2r)

h

10-4 Surface Area of Prisms and Cylinders

SURFACE AREA OF A CYLINDER

The surface area S of a cylinder is the sum of the areas of its bases, 2r2, plus the area of its lateral surface, 2rh.

S= 2r2 + 2rh

10-4 Surface Area of Prisms and Cylinders

Find the surface area of the cylinder formed by the net to the nearest tenth. Use 3.14 for .

Additional Example 2: Finding the Surface Area of a Cylinder

6 ft

6 ft

8.3 ft

S = 2r2 + 2rh

S (2 · 3.14 · 62) + (2 · 3.14 · 6 · 8.3) Substitute.

Use the formula.

S 226.08 + 312.744

S 538.824

S 538.8

The surface area of the cylinder is about 538.8 ft2.

Multiply.

Add.

Round.

10-4 Surface Area of Prisms and Cylinders

Find the surface area of the cylinder formed by the net to the nearest tenth. Use 3.14 for . 9 ft

9 ft

20 ft

S = 2r2 + 2rh

S (2 · 3.14 · 92) + (2 · 3.14 · 9 · 20) Substitute.

Use the formula.

S 508.68 + 1130.4

S 1,639.08

S 1,639.1

The surface area of the cylinder is about 1,639.1 ft2.

Multiply.

Add.

Round.

Check It Out: Example 2

10-4 Surface Area of Prisms and Cylinders

Course 2

Additional Example 3: Problem Solving Application

What percent of the total surface area of the soup can is covered by the label?

Soup

2 in.

5 in.6 in.

10-4 Surface Area of Prisms and Cylinders

Course 2

Additional Example 3 Continued

11 Understand the Problem

Rewrite the question as a statement.• Find the area of the label, and compare to the

total surface area of the can.

List the important information:• The can is approximately cylinder shaped.• The height for the can is 6 inches.• The radius of the can is 2 inches.• The height of the label is 5 inches.

10-4 Surface Area of Prisms and Cylinders

Course 2

Additional Example 3 Continued

Find the surface area of the can and the area of the label. Divide to find the percent of the surface area covered by the label.

22 Make a Plan

Soup6 in.

5 in.

2 in.

10-4 Surface Area of Prisms and Cylinders

Course 2

Additional Example 3 Continued

Solve33S = 2r2 + 2rh

2(3.14)(2)2 + 2(3.14)(2)(6) Substitute for r and h.

100.48 in.2

A = lw = (2r)w Substitute 2r for l . (2)(3.14)(2)(5) Substitute for r and w.

62.8 cm2

Percent of the surface covered by the label:

= 62.5 %. About 62.5% of the can’s surface is covered by the label.

62.8 cm2

100.48 cm2

10-4 Surface Area of Prisms and Cylinders

Course 2

Additional Example 3 Continued

Look Back44

Estimate the areas of the two rectangles in the net.

Label: 2(3)(2)(5) = 60 in2

Can: 2(3)(2)(6) = 77 in2

60 in.2 77 in.2

= 78%

The answer should be less than 78% because you did not consider the area of the two circles. So 62.5% is reasonable.

10-4 Surface Area of Prisms and Cylinders

Course 2

Check it Out: Example 3

What percent of the total surface area of the oil can is covered by the label?

6 in.

4 in.10 in.

10-4 Surface Area of Prisms and Cylinders

Course 2

Check It Out: Example 3 Continued

11 Understand the Problem

Rewrite the question as a statement.• Find the area of the label, and compare to the

total surface area of the can.

List the important information:• The can is approximately cylinder shaped.• The height for the can is 10 inches. • The diameter of the can is 6 inches.• The height of the label is 4 inches.

10-4 Surface Area of Prisms and Cylinders

Course 2

Check It Out: Example 3 Continued

Find the surface area of the can and the area of the label. Divide to find the percent of the surface area covered by the label.

22 Make a Plan

10 in. 4 in.

6 in.

10-4 Surface Area of Prisms and Cylinders

Course 2

Check It Out: Example 3 Continued

Solve33S = 2r2 + 2rh

2(3.14)(3)2 + 2(3.14)(3)(10) Substitute for r and h.

244.92 in2

A = lw = (2r)w Substitute 2r for l . (2)(3.14)(3)(4) Substitute for r and w.

75.36 cm2

Percent of the surface covered by the label:

= 30.7 %. About 30.7% of the can’s surface is covered by the label.

75.36 in.2

244.92 in.2

10-4 Surface Area of Prisms and Cylinders

Course 2

Check It Out: Example 3 Continued

Look Back44

Estimate the areas of the two rectangles in the net.

Label: 2(3)(3)(4) = 72 in2

Can: 2(3)(3)(10) = 180 in2

72 in.2 180 in.2

= 40%

The answer should be less than 40% because you did not consider the area of the two circles. So 30.7% is reasonable.

10-4 Surface Area of Prisms and Cylinders

Lesson QuizFind the surface area of each figure to the nearest tenth.

100.5 ft2352 ft2

25%

3. A drum is cylindrical, and its 14 in. width fits into a drum stand. What percent of the total surface area of the drum is covered by the 3 in. red stripe? Use 3.14 for

1. 2.