1. 10-4 Surface Areas of Prisms and Cylinders Warm Up Problem of the Day
Lesson Presentation
Course 2

2. Warm Up
Find 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
2. cone with radius 2 ft and height 3 ft
12.6 ft3
3. triangular pyramid with base area of 54 ft2 and height 9 ft
162 ft3

3. 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

4. Learn to find the surface area of prisms and cylinders.

5. Vocabulary
net
surface area

6. 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.

7. You can use nets to write formulas for the surface area of prisms
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

8. 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

9. Additional Example 1: Finding the Surface Area of a Prism
Find the surface area of the prism formed by the net. S
= 2lw + 2lh + 2wh S
= (2 · 15 · 9) + (2 · 15 · 7) + (2 · 9 · 7)
Substitute.
S =
Multiply.
S = 606
Add.
The surface area of the prism is 606 in2.

10. Find the surface area of the prism formed by the net.
Check It Out: Example 1
4 in.
Find the surface area of the prism formed by the net.
6 in.
3 in.
3 in.
4 in. S
= 2lw + 2lh + 2wh S
= (2 · 4 · 6) + (2 · 4 · 3) + (2 · 6 · 3)
Substitute.
S =
Multiply.
S = 108
Add.
The surface area of the prism is 108 in2.

11. 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, 2r, of the cylinder. So the area of the lateral surface is 2rh. The area of each base is r2. r
Circumference
of cylinder (2r) h

12. SURFACE AREA OF A CYLINDER
The surface area S of a cylinder is the sum of the areas of its bases, 2r2, plus the area of its lateral surface, 2rh.
S= 2r2 + 2rh

13. Additional Example 2: Finding the Surface Area of a Cylinder
Find the surface area of the cylinder formed by the net to the nearest tenth. Use 3.14 for .
6 ft
8.3 ft
6 ft
S = 2r2 + 2rh
Use the formula.
S  (2 · 3.14 · 62) + (2 · 3.14 · 6 · 8.3)
Substitute.
S 
Multiply.
S 
Add.
S  538.8
Round.
The surface area of the cylinder is about ft2.

14. Check It Out: Example 2
Find the surface area of the cylinder formed by the net to the nearest tenth. Use 3.14 for .
9 ft
20 ft
9 ft
S = 2r2 + 2rh
Use the formula.
S  (2 · 3.14 · 92) + (2 · 3.14 · 9 · 20)
Substitute.
S 
Multiply.
S  1,639.08
Add.
S  1,639.1
Round.
The surface area of the cylinder is about 1,639.1 ft2.

15. Additional Example 3: Problem Solving Application
What percent of the total surface area of the soup can is covered by the label?
2 in.
Soup
6 in.
5 in.
Course 2

16. Understand the Problem
Additional Example 3 Continued 1
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.
Course 2

17. Additional Example 3 Continued2
Make a Plan
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.
Soup
6 in.
5 in.
2 in.
Course 2

18. Additional Example 3 Continued
Solve 3
S = 2r2 + 2rh
 2(3.14)(2)2 + 2(3.14)(2)(6)
Substitute for r and h.
 in.2
A = lw
Substitute 2r for l .
= (2r)w
 (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 cm cm2
Course 2

19. Additional Example 3 Continued4
Look Back
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.
Course 2

20. Check it Out: Example 3
What percent of the total surface area of the oil can is covered by the label?
6 in.
10 in.
4 in.
Course 2

21. Understand the Problem
Check It Out: Example 3 Continued 1
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.
Course 2

22. Check It Out: Example 3 Continued2
Make a Plan
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.
10 in.
4 in.
6 in.
Course 2

23. Check It Out: Example 3 Continued
Solve 3
S = 2r2 + 2rh
Substitute for r and h.
 2(3.14)(3)2 + 2(3.14)(3)(10)
 in2
A = lw
Substitute 2r for l .
= (2r)w
 (2)(3.14)(3)(4)
Substitute for r and w.
 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 in.2
Course 2

24. Check It Out: Example 3 Continued4
Look Back
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 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.
Course 2

25. Lesson Quiz
Find the surface area of each figure to the nearest tenth. 1. 2.
100.5 ft2
352 ft2
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 p.
25%