1. 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.

2. 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.

3. The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism. The base of the rectangle is equal to the
perimeter of the base of the prism.

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

5. The surface area formula is only true for right prisms
The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces.
Caution!

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. Example 1B: Finding Lateral Areas and Surface Areas of Prisms
Find the lateral area and surface area of a right regular triangular prism with height 20 cm and base edges of length 10 cm. Round to the nearest tenth, if necessary.
L = Ph
= 30(20) = 600 ft2
P = 3(10) = 30 cm
S = Ph + 2B
The base area is

8. 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

9. 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.

11. 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

12. Example 2B: Finding Lateral Areas and Surface Areas of Right Cylinders
Find the lateral area and surface area of a right cylinder with circumference 24 cm and a height equal to half the radius. Give your answers in terms of .
Step 1 Use the circumference to find the radius.
C = 2r
Circumference of a circle
24 = 2r
Substitute 24 for C.
r = 12
Divide both sides by 2.

13. Example 2B Continued
Find the lateral area and surface area of a right cylinder with circumference 24 cm and a height equal to half the radius. Give your answers in terms of .
Step 2 Use the radius to find the lateral area and surface area. The height is half the radius, or 6 cm.
L = 2rh = 2(12)(6) = 144 cm2
Lateral area
S = L + 2r2 = 144 + 2(12)2
= 432 in2
Surface area

14. 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

15. 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

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

17. 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.

18. 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

19. Check It Out! Example 3
Find the surface area of the composite figure. Round to the nearest tenth.

20. Check It Out! Example 3 Continued
Find the surface area of the composite figure. Round to the nearest tenth.
The surface area of the rectangular prism is
S =Ph + 2B = 26(5) + 2(36) = 202 cm2.
The surface area of the cylinder is
S =Ph + 2B = 2(2)(3) + 2(2)2 = 20 ≈ 62.8 cm2.
The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure.

21. Check It Out! Example 3 Continued
Find the surface area of the composite figure. Round to the nearest tenth.
S = (rectangular surface area) +
(cylinder surface area) – 2(cylinder base area)
S = — 2()(22) = cm2

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

23. 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.