1. Warm Up
Find the perimeter and area of
each polygon.
1. a rectangle with base 14 cm and height 9 cm
2. a right triangle with 9 cm and 12 cm legs
3. an equilateral triangle with side length 6 cm
P = 46 cm; A = 126 cm2
P = 36 cm; A = 54 cm2

2. Objectives
Learn and apply the formula for the surface area of a prism.
Learn and apply the formula for the surface area of a cylinder.

3. Prisms and cylinders have 2 congruent parallel bases.
A lateral face is not a base.
The edges of the base are called base edges.
A lateral edge is not an edge of a base.
The lateral faces of a right prism are all rectangles.
An oblique prism has at least one nonrectangular lateral face.

4. An altitude of a prism or cylinder is a perpendicular segment joining the planes of the bases.
The height of a three-dimensional figure is the length of an altitude.
Surface area is the total area of all faces and curved
surfaces of a three-dimensional figure.
The lateral area of a prism is the sum of the areas of the lateral faces.

5. The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as
S = 2ℓw + 2wh + 2ℓh.

6. Example 1A: Finding Lateral Areas and Surface Areas of Prisms
Find the lateral area and surface area of the right rectangular prism. Round to the nearest tenth, if necessary.
L = Ph
P = 2(9) + 2(7) = 32 ft
= 32(14) = 448 ft2
S = Ph + 2B
= (7)(9) = 574 ft2

7. Check It Out! Example 1
Find the lateral area and surface area of a cube with edge length 8 cm.
L = Ph
= 32(8) = 256 cm2
P = 4(8) = 32 cm
S = Ph + 2B
= (8)(8) = 384 cm2

8. The lateral surface of a cylinder is the curved surface that connects the two bases.
The axis of a cylinder is the segment with endpoints at the centers of the bases.
The axis of a right cylinder is perpendicular to its bases.
The axis of an oblique cylinder is not perpendicular to its bases.
The altitude of a right cylinder is the same length as the axis.

10. Example 2A: Finding Lateral Areas and Surface Areas of Right Cylinders
Find the lateral area and surface area of the right cylinder. Give your answers in terms of .
The radius is half the diameter, or 8 ft.
L = 2rh = 2(8)(10) = 160 in2
S = L + 2r2 = 160 + 2(8)2
= 288 in2

11. Check It Out! Example 2
Find the lateral area and surface area of a cylinder with a base area of 49 and a height that is 2 times the radius.
Step 1 Use the circumference to find the radius.
A = r2
Area of a circle
49 = r2
Substitute 49 for A.
Divide both sides by  and take the square root.
r = 7

12. Check It Out! Example 2 Continued
Find the lateral area and surface area of a cylinder with a base area of 49 and a height that is 2 times the radius.
Step 2 Use the radius to find the lateral area and surface area. The height is twice the radius, or 14 cm.
L = 2rh = 2(7)(14)=196 in2
Lateral area
S = L + 2r2 = 196 + 2(7)2 =294 in2
Surface area

13. Example 3: Finding Surface Areas of Composite Three-Dimensional Figures
Find the surface area of the composite figure.

14. Example 3 Continued
The surface area of the rectangular prism is .
A right triangular prism is added to the rectangular prism. The surface area of the triangular prism is .
Two copies of the rectangular prism base are removed. The area of the base is B = 2(4) = 8 cm2.

15. Example 3 Continued
The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure.
S = (rectangular prism surface area) + (triangular prism surface area) – 2(rectangular prism base area)
S = – 2(8) = 72 cm2

16. Example 4: Exploring Effects of Changing Dimensions
The edge length of the cube is tripled. Describe the effect on the surface area.

17. Example 4 Continued original dimensions: edge length tripled: S = 6ℓ2
24 cm
original dimensions:
edge length tripled:
S = 6ℓ2
S = 6ℓ2
= 6(8)2 = 384 cm2
= 6(24)2 = 3456 cm2
Notice than 3456 = 9(384). If the length, width, and height are tripled, the surface area is multiplied by 32, or 9.

18. Check It Out! Example 5
A piece of ice shaped like a 5 cm by 5 cm by 1 cm rectangular prism has approximately the same volume as the pieces below. Compare the surface areas. Which will melt faster?
The 5 cm by 5 cm by 1 cm prism has a surface area of 70 cm2, which is greater than the 2 cm by 3 cm by
4 cm prism and about the same as the half cylinder. It will melt at about the same rate as the half cylinder.

19. Lesson Quiz: Part I
Find the lateral area and the surface area of each figure. Round to the nearest tenth, if necessary.
1. a cube with edge length 10 cm
2. a regular hexagonal prism with height 15 in. and base edge length 8 in.
3. a right cylinder with base area 144 cm2 and a height that is the radius
L = 400 cm2 ; S = 600 cm2
L = 720 in2; S  in2
L  cm2; S = cm2

20. Lesson Quiz: Part II
4. A cube has edge length 12 cm. If the edge length of the cube is doubled, what happens to the surface area?
5. Find the surface area of the composite figure.
The surface area is multiplied by 4.
S = 3752 m2

21. Homework
Worksheet 10-4