1. Surface Areas of Prisms, Cylinders, Pyramids and Cones
Objective – Find the surface area of prisms and cylinders.

2. Students compute the surface areas of prisms, pyramids, cylinders, and cones; and students commit to memory the formulas for prisms, pyramids, and cylinders.

3. Types of Prisms Lateral Edges Lateral Faces Bases Bases
Triangular Prism
Pentagonal Prism h h
Right Prism
Oblique Prism

4. Key Concept(s): Lateral Area and Surface Area of a Prism
Base Perimeter h
Lateral Area
Base Perimeter
Height
Base Area
Surface Area
Lateral Area
Area of the Base(s)

5. #1 Finding the Surface Area of a Prism
Find the surface area of the regular hexagonal prism.
P = 36
12 m
6 m

6. #2 Finding the Surface Area of a Prism
Find the surface area of the triangular prism.
12 cm
5 cm
6 cm
P = 16
5 cm

7. Key Concept(s): Lateral Area and Surface Area of a Cylinder Base Area
Radius
Lateral Area
Radius
Height h
Surface Area
Lateral Area
Base Area

8. #3 Finding the Surface Area of a Cylinder
Find the surface area of a cylinder with height 10 cm and radius 5 cm in terms of pi.
5 cm
10 cm

9. #4 Finding the Surface Area of a Cylinder
Find the surface area of a cylinder with radius 6 ft and height 9 ft in terms of pi.
9 ft
6 ft

10. Example:
Find the surface area of a cylinder with radius 3 ft and height 6 ft.
S.A. = 2πrh + 2B
7 ft
5 ft

11. Example: Find the surface area of the triangular prism. S.A. = ph + 2B
10 cm
5 cm
4 cm
7 cm

12. Surface Areas of Pyramids

13. Pyramids
Pyramid – A three dimensional figure in which one face, the base, is any polygon and the lateral faces are triangles that meet at a common vertex.
vertex
Pyramid

14. Pyramids cont.
The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.
The length of the altitude is called the height.
Height
Pyramid

15. Pyramids cont.
Regular Pyramid – a pyramid whose base is a regular polygon.
Slant Height – the length of the altitude of the lateral face.
Slant Height
Pyramid

16. Formulas Lateral Area and Surface Area of a Regular Pyramid
Base Perimeter
Slant Height B
Surface Area
Lateral Area
Base Area

17. Finding Surface Area of a Pyramid
Find the surface area of a square pyramid with base edges 5 m and slant height 3 m.

18. #2 Surface Area of a Pyramid
Find the surface area of a regular hexagonal pyramid with a slant height 20 in and sides 8 in.

19. #3 Find the surface area of:
12 m

20. #4 Find the surface area of:
10”
15”
S = ½ (40)(15) + (10)(10)
S = 400 in2

21. CIRCULAR BASE
A cone is in the form of a pyramid, but has a circle as its base.
a. Convert the formula you created in the previous problem to find the surface area of a cone.

22. b. Find the surface area of the following cones.

25. Your class has decided to throw your principal a surprise birthday party. The whole class is working together to create party decorations, and your team has been assigned the job of producing party hats. Each party hat will be created out of special decorative paper and will be in the shape of a cone.

26. Your Task: Use the sample party hat provided by your teacher to determine the size and shape of the paper that forms the hat. Then determine the amount of paper (in square centimeters) needed to produce one party hat and figure out the total amount of paper you will need for each person in your class to have a party hat.
Lateral area of each cone

28. The vertex of a pyramid is the point opposite the base of the pyramid
The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.

29. The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid.

31. Finding Lateral Area and Surface Area of Pyramids
Find the lateral area and surface area of a regular square pyramid with base edge length 14 cm and slant height 25 cm. Round to the nearest tenth, if necessary.
Lateral area of a regular pyramid
P = 4(14) = 56 cm
Surface area of a regular pyramid
B = 142 = 196 cm2

32. Finding Lateral Area and Surface Area of Pyramids
Find the lateral area and surface area of the regular pyramid.
Step 1 Find the base perimeter and apothem.
The base perimeter is 6(10) = 60 in.
The apothem is so the base area is

33. Example 1B Continued
Find the lateral area and surface area of the regular pyramid.
Step 2 Find the lateral area.
Lateral area of a regular pyramid
Substitute 60 for P and 16 for ℓ.

34. Example 1B Continued
Find the lateral area and surface area of the regular pyramid.
Step 3 Find the surface area.
Surface area of a regular pyramid
Substitute for B.

35. Check It Out! Example 1
Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft.
Step 1 Find the base perimeter and apothem. The base perimeter is 3(6) = 18 ft.
The apothem is so the base area is

36. Check It Out! Example 1 Continued
Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft.
Step 2 Find the lateral area.
Lateral area of a regular pyramid
Substitute 18 for P and 10 for ℓ.

37. Check It Out! Example 1 Continued
Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10 ft.
Step 3 Find the surface area.
Surface area of a regular pyramid
Substitute for B.

38. The vertex of a cone is the point opposite the base
The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base.

39. The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular
segment from the vertex of the cone to the plane of the base.

40. Finding Lateral Area and Surface Area of Right Cones
Find the lateral area and surface area of a right cone with radius 9 cm and slant height 5 cm.
L = rℓ
Lateral area of a cone
= (9)(5) = 45 cm2
Substitute 9 for r and 5 for ℓ.
S = rℓ + r2
Surface area of a cone
= 45 + (9)2 = 126 cm2
Substitute 5 for ℓ and 9 for r.

41. Finding Lateral Area and Surface Area of Right Cones
Find the lateral area and surface area of the cone.
Use the Pythagorean Theorem to find ℓ.
L = rℓ
Lateral area of a right cone
= (8)(17)
= 136 in2
Substitute 8 for r and 17 for ℓ.
S = rℓ + r2
Surface area of a cone
= 136 + (8)2
= 200 in2
Substitute 8 for r and 17 for ℓ.

42. Check It Out! Example 2
Find the lateral area and surface area of the right cone.
Use the Pythagorean Theorem to find ℓ. ℓ
L = rℓ
Lateral area of a right cone
= (8)(10)
= 80 cm2
Substitute 8 for r and 10 for ℓ.
S = rℓ + r2
Surface area of a cone
= 80 + (8)2
= 144 cm2
Substitute 8 for r and 10 for ℓ.

43. Finding Surface Area of Composite Figures
Find the surface area of the composite figure.
Left-hand cone:
The lateral area of the cone is
L = rl = (6)(12) = 72 in2.
Right-hand cone:
Using the Pythagorean Theorem, l = 10 in.
The lateral area of the cone is
L = rl = (6)(10) = 60 in2.

44. Example 4 Continued
Find the surface area of the composite figure.
Composite figure: 
S = (left cone lateral area) + (right cone lateral area)
= 60 in2 + 72 in2 = 132 in2