1. 11.6 Surface Area and Volume of Pyramids and Cones
Geometry
11.6 Surface Area and Volume of Pyramids and Cones

2. Geometry 12.3 Surface Area of Pyramids and Cones
11.6 Essential Question
How do you find the surface area and volume of a pyramid or a cone?
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones

3. Geometry 12.3 Surface Area of Pyramids and Cones
Goals
Know what a pyramid is.
Find the surface area and volume of a pyramid.
Know what a cone is.
Find the surface area and volume of a cone.
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Geometry 12.3 Surface Area of Pyramids and Cones

4. Geometry 12.3 Surface Area of Pyramids and Cones
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Geometry 12.3 Surface Area of Pyramids and Cones

5. Geometry 12.3 Surface Area of Pyramids and Cones
Polyhedron with one polygon base.
Lateral faces are triangles.
Lateral faces meet at one common vertex.
Regular pyramid: base is a regular polygon. (This is what we will discuss.)
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Geometry 12.3 Surface Area of Pyramids and Cones

6. Geometry 12.3 Surface Area of Pyramids and Cones
Square Pyramid
Vertex
Slant Height (l)
Altitude (h)
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Geometry 12.3 Surface Area of Pyramids and Cones

7. Surface Area of a Pyramid
The surface area is the sum of the area of the base and the area of the lateral faces.
SA = B + LA
Area of Base
Area of Lateral Sides
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Geometry 12.3 Surface Area of Pyramids and Cones

8. Surface Area of a Pyramid
Unfold the pyramid:
The surface area is the sum of the area of the base and the area of the lateral faces.
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Geometry 12.3 Surface Area of Pyramids and Cones

9. Geometry 12.3 Surface Area of Pyramids and Cones
Lateral Area
Re-arrange the triangles.
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Geometry 12.3 Surface Area of Pyramids and Cones

10. Geometry 12.3 Surface Area of Pyramids and Cones
Lateral Area ? l S S S S ?
Find the area of one triangle. S l
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Geometry 12.3 Surface Area of Pyramids and Cones

11. Geometry 12.3 Surface Area of Pyramids and Cones
Lateral Area l S S S S
Find the area of four triangles.
What is 4s?
Perimeter!
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones

12. Lateral Area of a Pyramid
𝑃=𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑡ℎ𝑒 𝐵𝑎𝑠𝑒
𝑙=𝑡ℎ𝑒 𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡
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Geometry 12.3 Surface Area of Pyramids and Cones

13. Surface Area of a Pyramid
Just add the area of the base to the LA. S l
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones

14. Geometry 12.5 Volume of Pyramids and Cones
Volume of a Pyramid
Volume of the Prism
V = Bh
Volume of the Pyramid
𝐵=𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝐵𝑎𝑠𝑒
ℎ=ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑦𝑟𝑎𝑚𝑖𝑑
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Geometry 12.5 Volume of Pyramids and Cones

15. Example 1 Name the solid. Then find the SA and V.12 10
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Geometry 12.3 Surface Area of Pyramids and Cones

16. Finding the slant height.12 5 10
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Geometry 12.3 Surface Area of Pyramids and Cones

17. Finding the surface area.10 12 5 13
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Geometry 12.3 Surface Area of Pyramids and Cones

18. Example 2 Name the solid. Then find the SA and V.14 25
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Geometry 12.3 Surface Area of Pyramids and Cones

19. Example 3 Name the solid. Then find the SA and V.12 10 10
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Geometry 12.5 Volume of Pyramids and Cones

20. Geometry 12.5 Volume of Pyramids and Cones
Example 3 Solution ? h 12 5 10 10
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Geometry 12.5 Volume of Pyramids and Cones

21. Geometry 12.5 Volume of Pyramids and Cones
Example 3 Solution
Solve for h:
10.91 h 12 5 10 10
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Geometry 12.5 Volume of Pyramids and Cones

22. Geometry 12.5 Volume of Pyramids and Cones
Example 3 Solution
Solve for V:
10.91 12 5 10 10
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Geometry 12.5 Volume of Pyramids and Cones

23. Example 4 Name the solid. Then find the SA and V.6
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Geometry 12.5 Volume of Pyramids and Cones

24. Geometry 12.5 Volume of Pyramids and Cones
Example 4 Solution
Area of the base:
h = 15 6 6 3
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Geometry 12.5 Volume of Pyramids and Cones

25. Geometry 12.5 Volume of Pyramids and Cones
Example 4 Solution
Area of the base:
h = 15 6
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Geometry 12.5 Volume of Pyramids and Cones

26. Geometry 12.3 Surface Area of Pyramids and Cones
Circular base
No Lateral Faces
SA = B + LA
Can be right or oblique
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Geometry 12.3 Surface Area of Pyramids and Cones

27. Geometry 12.3 Surface Area of Pyramids and Cones
Cone Surface Area
You get:
Circular Base
Cut the cone from the vertex to the base.
Then around the base.
Circle Sector
1st
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Geometry 12.3 Surface Area of Pyramids and Cones

28. Geometry 12.3 Surface Area of Pyramids and Cones
Cone Surface Area
A = r2 l
Circular Base r
2r
The length of this arc is the circumference of the base. l
Circle Sector
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones

29. Geometry 12.3 Surface Area of Pyramids and Cones
Sector Area
2r l
Circle Sector
This is the lateral area of a cone.
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones

30. Geometry 12.3 Surface Area of Pyramids and Cones
Cone Surface Area
A = r2 l
Circular Base r
2r
A = rl
SA = r2 + rl l
Circle Sector
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones

31. Geometry 12.3 Surface Area of Pyramids and Cones
Cone Surface Area
SA = r2 + rl
LA = rl l h r
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones

32. Geometry 12.5 Volume of Pyramids and Cones
Volume of a Cone
Volume of the Cylinder
V = r2h
Volume of the Cone
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Geometry 12.5 Volume of Pyramids and Cones

33. Example 5 Find the SA & V of the right cone.12
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Geometry 12.5 Volume of Pyramids and Cones

34. Example 6 Find theSA & V of the right cone.15 6
3rd
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Geometry 12.5 Volume of Pyramids and Cones

35. Geometry 12.3 Surface Area of Pyramids and Cones
Now you can find the surface area and volume of cone or a pyramid!
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones

36. Geometry 12.5 Volume of Pyramids and Cones
Summary
The volume of a pyramid is 1/3 the volume of a prism with the same measurements.
The volume of a cone is 1/3 the volume of a cylinder with the same measurements.
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Geometry 12.5 Volume of Pyramids and Cones

37. Geometry 12.3 Surface Area of Pyramids and Cones
Summary - Pyramid h l S
These should be on your formula sheet.
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones

38. Geometry 12.5 Volume of Pyramids and Cones
Summary - Cone r h l
LA = rl
SA = r2 + rl
Add these to your formula sheet.
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Geometry 12.5 Volume of Pyramids and Cones

39. Problem 1 Which figure has the larger volume?12 14 14 12
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Geometry 12.5 Volume of Pyramids and Cones

40. Geometry 12.5 Volume of Pyramids and Cones
Problem 2
The base of a square pyramid measures 12 ft on each side, and the height of the pyramid is 15 ft.
A cone has a diameter of 12 ft and has the same volume as the square pyramid. What is the height of the cone?
November 23, 2018
Geometry 12.5 Volume of Pyramids and Cones

41. Geometry 12.5 Volume of Pyramids and Cones
Solution
Cone
Pyramid
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Geometry 12.5 Volume of Pyramids and Cones

42. Geometry 12.3 Surface Area of Pyramids and Cones
Problem 3
Find the volume of the composite figure.
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Geometry 12.3 Surface Area of Pyramids and Cones

43. Geometry 12.3 Surface Area of Pyramids and Cones
Problem 4
Find the volume of the composite figure.
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Geometry 12.3 Surface Area of Pyramids and Cones

44. Geometry 12.3 Surface Area of Pyramids and Cones
Homework
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Geometry 12.3 Surface Area of Pyramids and Cones

45. Geometry 12.3 Surface Area of Pyramids and Cones
Cone Storage
November 23, 2018
Geometry 12.3 Surface Area of Pyramids and Cones