1. Entry Task

2. I can apply the formula for the surface area of a pyramid and cone. Learning Target Surface Areas of Pyramids Success Criteria: I can apply the formula for the surface area of a pyramid and cone to real world problems.

3. Pyramids Pyramid 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

4. 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

5. 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

6. Summary….

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

8. 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.

9. Example 1: Finding Surface Area of a Pyramid Find the surface area of a square pyramid with base edges 5 m and slant height 3 m.

10. #2 Finding Surface Area of a Pyramid Find the surface area of a regular hexagonal pyramid with a slant height 20 in and sides 8 in.

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

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

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

14. 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.

15. 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. B

16. Example 1: Find the surface area of the cone to the nearest whole number. a.r = 4 slant height = 6 4 in. 6 in.

17. Example 2: Find the surface area of the cone to the nearest whole number. b. First, find the slant height. Next, r = 12, 12 ft. 5 ft.

18. Example 2B: 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 =  r ℓ =  (8)(17) = 136  in 2 Lateral area of a right cone Substitute 8 for r and 17 for ℓ. S =  r ℓ +  r 2 Surface area of a cone = 136  +  (8) 2 = 200  in 2 Substitute 8 for r and 17 for ℓ.

19. On your own #1 Calculate the surface area of: S =  (7) 2 +  (7)(11.40) S = 49  + 79.80  S = 128.8 

20. Homework p. 713 #9,10,12,13,15,16,17,19,21,22

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

22. Example 1A: 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 = 14 2 = 196 cm 2

23. 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

24. 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.

25. Step 3 Find the surface area. Surface area of a regular pyramid 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. Substitute for B.

26. Example 2A: 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  cm 2 Substitute 9 for r and 5 for ℓ. S =  rℓ +  r 2 Surface area of a cone = 45  +  (9) 2 = 126  cm 2 Substitute 5 for ℓ and 9 for r.

27. Example 3: Exploring Effects of Changing Dimensions The base edge length and slant height of the regular hexagonal pyramid are both divided by 5. Describe the effect on the surface area.

28. 3 in. 2 in. Example 3 Continued original dimensions: base edge length and slant height divided by 5: S = Pℓ + B 1212 1212

29. Example 3 Continued original dimensions: base edge length and slant height divided by 5: 3 in. 2 in. Notice that. If the base edge length and slant height are divided by 5, the surface area is divided by 5 2, or 25.

30. Check It Out! Example 3 The base edge length and slant height of the regular square pyramid are both multiplied by. Describe the effect on the surface area.

31. Check It Out! Example 3 Continued original dimensions:multiplied by two-thirds: By multiplying the dimensions by two-thirds, the surface area was multiplied by. 8 ft 10 ft S = Pℓ + B 1212 = 260 cm 2 S = Pℓ + B 1212 = 585 cm 2

32. Example 4: Finding Surface Area of Composite Three-Dimensional Figures Find the surface area of the composite figure. The lateral area of the cone is L =  r l =  (6)(12) = 72  in 2. Left-hand cone: Right-hand cone: Using the Pythagorean Theorem, l = 10 in. The lateral area of the cone is L =  r l =  (6)(10) = 60  in 2.

33. Example 4 Continued Composite figure: S = (left cone lateral area) + (right cone lateral area) Find the surface area of the composite figure. = 60  in 2 + 72  in 2 = 132  in 2

34. Check It Out! Example 4 Find the surface area of the composite figure. Surface Area of Cube without the top side: S = 4wh + B S = 4(2)(2) + (2)(2) = 20 yd 2

35. Check It Out! Example 4 Continued Surface Area of Pyramid without base: Surface Area of Composite: Surface of Composite = SA of Cube + SA of Pyramid