Geometry Surface Area and Volume of Pyramids and Cones.

Transcription

1 Geometry 11.6 Surface Area and Volume of Pyramids and Cones

2 11.6 Essential Question How do you find the surface area and volume of a pyramid or a cone? Geometry 1.3 Surface Area of Pyramids and Cones

3 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. Geometry 1.3 Surface Area of Pyramids and Cones 3

4 Pyramids Geometry 1.3 Surface Area of Pyramids and Cones 4

5 Pyramid 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.) Geometry 1.3 Surface Area of Pyramids and Cones 5

6 Square Pyramid Vertex Altitude (h) Slant Height (l) Geometry 1.3 Surface Area of Pyramids and Cones 6

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 Geometry 1.3 Surface Area of Pyramids and Cones 7

8 Surface Area of a Pyramid S Unfold the pyramid: The surface area is the sum of the area of the base and the area of the lateral faces. Geometry 1.3 Surface Area of Pyramids and Cones 8

9 Lateral Area Re-arrange the triangles. Geometry 1.3 Surface Area of Pyramids and Cones 9

10 Lateral Area? l? S S S S Find the area of one triangle. A 1 sl l S Geometry 1.3 Surface Area of Pyramids and Cones 10

11 Lateral Area l S S S S Find the area of four triangles. A LA 4 1 sl What is 4s? 1 (4 sl ) Perimeter! Geometry 1.3 Surface Area of Pyramids and Cones 11

12 Lateral Area of a Pyramid LA 1 Pl l P = Perimeter of the Base l = the slant height Geometry 1.3 Surface Area of Pyramids and Cones 1

13 Surface Area of a Pyramid Just add the area of the base to the LA. l S LA 1 Pl SA B Pl 1 Geometry 1.3 Surface Area of Pyramids and Cones 13

14 Volume of a Pyramid Volume of the Prism V = Bh Volume of the Pyramid V 1 Bh 3 B = Area of the Base h = height of the pyramid Geometry 1.5 Volume of Pyramids and Cones 14

15 Example 1 Name the solid. Then find the SA and V Geometry 1.3 Surface Area of Pyramids and Cones 15

16 Finding the slant height. l l Geometry 1.3 Surface Area of Pyramids and Cones 16

17 Finding the surface area SA B PL Geometry 1.3 Surface Area of Pyramids and Cones 17

18 Example Name the solid. Then find the SA and V Geometry 1.3 Surface Area of Pyramids and Cones 18

19 Example 3 find the SA and V. Name the solid. Then Geometry 1.5 Volume of Pyramids and Cones 19

20 Example 3 Solution? h Geometry 1.5 Volume of Pyramids and Cones 0

21 Example 3 Solution Solve for h: h h h h h h Geometry 1.5 Volume of Pyramids and Cones 1

22 Example 3 Solution Solve for V: 10 V Bh Geometry 1.5 Volume of Pyramids and Cones

23 Example 4 h = 15 Name the solid. Then find the SA and V. l = Geometry 1.5 Volume of Pyramids and Cones 3

24 Example 4 Solution Area of the base: h = A 3 1 ap Geometry 1.5 Volume of Pyramids and Cones 4

25 Example 4 h = 15 Solution Area of the base: V Bh Geometry 1.5 Volume of Pyramids and Cones 5

26 Cones Circular base No Lateral Faces SA = B + LA Can be right or oblique Geometry 1.3 Surface Area of Pyramids and Cones 6

27 Cone Surface Area You get: Circular Base Cut the cone from the vertex to the base. 1 st Then around the base. Circle Sector Geometry 1.3 Surface Area of Pyramids and Cones 7

28 Cone Surface Area A = r l r r The length of this arc is the circumference of the base. Circular Base l Circle Sector Geometry 1.3 Surface Area of Pyramids and Cones 8

29 Sector Area r A l r l l Circle Sector r l A l A rl This is the lateral area of a cone. Geometry 1.3 Surface Area of Pyramids and Cones 9

30 Cone Surface Area A = r l r SA = r + rl r Circular Base A = rl l Circle Sector Geometry 1.3 Surface Area of Pyramids and Cones 30

31 Cone Surface Area h r l SA = r + rl LA = rl Geometry 1.3 Surface Area of Pyramids and Cones 31

32 Volume of a Cone Volume of the Cylinder V = r h Volume of the Cone V 1 3 r h Geometry 1.5 Volume of Pyramids and Cones 3

33 Example 5 Find the SA & V of the right cone. 1 5 Geometry 1.5 Volume of Pyramids and Cones 33

34 Example 6 Find thesa & V of the right cone rd Geometry 1.5 Volume of Pyramids and Cones 34

35 Now you can find the surface area and volume of cone or a pyramid! Geometry 1.3 Surface Area of Pyramids and Cones 35

36 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. Geometry 1.5 Volume of Pyramids and Cones 36

37 Summary - Pyramid LA 1 Pl 1 SA B Pl h l V 1 3 Bh S These should be on your formula sheet. Geometry 1.3 Surface Area of Pyramids and Cones 37

38 Summary - Cone LA = rl SA = r + rl h l V 1 3 r h r Add these to your formula sheet. Geometry 1.5 Volume of Pyramids and Cones 38

39 Problem 1 Which figure has the larger volume? Geometry 1.5 Volume of Pyramids and Cones 39

40 Problem The base of a square pyramid measures 1 ft on each side, and the height of the pyramid is 15 ft. A cone has a diameter of 1 ft and has the same volume as the square pyramid. What is the height of the cone? Geometry 1.5 Volume of Pyramids and Cones 40

41 Solution Cone V Pyramid Bh V h r h h h Geometry 1.5 Volume of Pyramids and Cones 41

42 Problem 3 Find the volume of the composite figure. Geometry 1.3 Surface Area of Pyramids and Cones 4

43 Problem 4 Find the volume of the composite figure. Geometry 1.3 Surface Area of Pyramids and Cones 43

44 Homework Geometry 1.3 Surface Area of Pyramids and Cones 44

45 Cone Storage Geometry 1.3 Surface Area of Pyramids and Cones 45