1 7th Grade Math 3D Geometry 20151120 www.njctl.org 2 Table of - PowerPoint PPT Presentation

Demonstration Comparing Volume of Cones & Spheres with Volume of Cylinders click to go to web site 52

Volume of a Cone (Area of Base x Height) 3 The Volume of a Cone is 1/3 the 1 click to reveal (Area of Base x Height) volume of a 3 cylinder with the same base area ( B ) and height ( h ). 53

Volume of a Sphere V = 2/3 (Volume of Cylinder) The Volume of a Sphere is 2/3 the ( ) 2/3 V= r 2 h click to reveal volume of a or cylinder with the V = 4/3 r 3 same base area ( B ) and height ( h ). 54

Volume How much ice cream can a Friendly’s Waffle cone hold if it has a diameter of 6 in and its height is 10 in? Answer (Just Ice Cream within Cone. Not on Top) 55

24 Find the volume. Answer 9 in 4 in 56

25 Find the volume. Answer 5 cm 8 cm 57

Volume If the radius of a sphere is 5.5 cm, what is its volume? 58

26 What is the volume of a sphere with a radius of 8 ft? Answer 59

27 What is the volume of a sphere with a diameter of 4.25 in? Answer 60

Volume of a Pyramid (Area of Base x Height) 3 The Volume of a 1 click to reveal (Area of Base x Height) Pyramid is 1/3 the 3 volume of a prism with the same base area ( B ) and height ( h ). Note: Pyramids are named by the shape of their base. 61

Volume Example: Find the volume of the pyramid shown below. 1 V = Bh 3 1 V = Bh 3 =5 m m 4 = h t g n e l e d i s 62

28 Find the Volume of a triangular pyramid with base edges of 8 in, base height of 6.9 in and a pyramid height of 10 in. Answer 10 in 6.9 in 8 in 63

29 Find the volume. Answer 15.3 cm 7 cm 8 cm 64

Surface Area Return to Table of Contents 65

Surface Area of Prisms Return to Table of Contents 66

Surface Area Surface Area is the sum of the areas of all outside surfaces of a 3­D figure. To find surface area, you must find the area of each surface of the figure then add them together. What type of figure is pictured? 6 in How many surfaces are there? 3 in How do you find the area of each 8 in surface? 67

Surface Area 6 in 3 in 8 in Bottom Top Left Right Front Back SUM 8 8 6 6 8 8 x 3 x 3 x 3 x 3 x 6 x 6 24 24 18 18 48 48 = 180 in 2 18 6 in 6 in 48 6 in +48 3 in 3 in 3 in 8 in 8 in 8 in 180 in 2 68

Net Another way that you can visualize the entire surface and calculate the surface area is to create the net of your solid by unfolding it. Below is a the net of a rectangular prism. Back Right Left Bottom Top Front 69

Surface Area using Nets You can also calculate the surface area of our last example by drawing the net, calculating the areas, and adding them together. 48 in 2 6 in 24 in 2 18 in 2 24 in 2 18 in 2 3 in 8 in 48 in 2 Total Surface Area: 18 + 24 + 18 + 24 + 48 + 48 = 180 in 2 70

Arrangement of Unit Cubes Surface Area Activity Click the link below for the activity Lab #2: Surface Area Activity 71

The next 4 slides address MP2 Teachers: Math Practice Ask: How can you represent the Use this Mathematical Practice Pull Tab for the next 4 problem w/ symbols and numbers? SMART Response slides. What do you think the answer/result will be? 72

30 Which arrangement of 27 cubes has the least surface area? A 1 x 1 x 27 B 3 x 3 x 3 Answer C 9 x 3 x 1 73

31 Which arrangement of 12 cubes has the least surface area? A 2 x 2 x 3 B 4 x 3 x 1 Answer C 2 x 6 x 1 D 1 x 1 x 12 74

32 Which arrangement of 25 cubes has the greatest surface area? A 1 x 1 x 25 B 1 x 5 x 5 Answer 75

33 Which arrangement of 48 cubes has the least surface area? A 4 x 12 x 1 B 2 x 2 x 12 C 1 x 1 x 48 Answer D 3 x 8 x 2 E 4 x 2 x 6 F 4 x 3 x 4 76

Surface Area Find the surface area of a rectangular shoe box that has a length of 12 inches, a width of 6 inches and a height of 5 inches. 5 in 6 in 12 in Top/Bottom Front/Back Left/Right Surface Area 12 12 6 144 x 6 x 5 x 5 120 72 60 30 + 60 x 2 x 2 x 2 324 in 2 144 120 60 click to reveal click to reveal click to reveal click to reveal click to reveal click to reveal 77

Surface Area Name the figure. Find the figure's surface area. Rectangular Prism Answer 7 m Base 4 Rectangles 3(5)(2) 3(7) 5(7) 3(7) 5(7) 30 21 35 35 21 Sum = 142 m 2 3 m 5 m Click to reveal the prism's net, if needed 78

34 How many faces does the figure have? Answer 6 m 2 m 4 m 79

35 How many area problems must you complete when finding the surface area? Answer 6 m 2 m 4 m 80

36 What is the area of the top or bottom face? Answer 6 m 2 m 4 m 81

37 What is the area of the left or right face? Answer 6 m 2 m 4 m 82

38 What is the area of the front or back face? Answer 6 m 2 m 4 m 83

39 What is the surface area of the figure? Answer 6 m 2 m 4 m Click to reveal the prism's net, if needed 84

Find the Surface Area 1. Draw and label ALL faces; use the net, if it's helpful 2. Find the correct dimensions for each face 3. Calculate the AREA of EACH face 4. Find the SUM of ALL faces 5. Label Answer 5 yd go on to see steps 5 yd 4 yd 9 yd 6 yd 85

Surface Area 5 yd 5 yd 5 yd 5 yd 4 yd 4 yd 9 yd 9 yd 6 yd 6 yd Triangles Bottom Rectangle 4 9 CLICK TO REVEAL x 6 x 6 CLICK TO REVEAL 24 / 2 = 12 54 x 2 24 Total Left/Right Rectangles Surface (Same size since isosceles) Area 5 yd CLICK TO REVEAL 24 5 5 yd 54 x 9 4 yd CLICK TO REVEAL 9 yd + 90 45 6 yd 168 yd 2 x 2 90 86

Find the Surface Area Using the Net 5 yd 5 yd 4 yd 9 yd 6 yd Triangles 4 Middle Rectangle x 6 CLICK TO REVEAL 9 24 / 2 = 12 Total CLICK TO REVEAL x 6 x 2 Surface 54 24 Area CLICK TO REVEAL 24 Left/Right Rectangles 54 CLICK TO REVEAL (Same size since isosceles) + 90 5 x 9 = 45 x 2 = 90 168 yd 2 87

Find the Surface Area 1. Draw and label ALL faces; use the net if it's helpful 2. Find the correct dimensions for each face 3. Calculate the AREA of EACH face 4. Find the SUM of ALL faces 5. Label Answer 9 cm TRY THIS ONE 9 cm 7.8 cm 11 cm 9 cm 88

Surface Area 9 cm 9 cm Answer 9 cm 9 cm 7.8 cm 11 cm 7.8 cm 11 cm 9 cm 9 cm Triangles Rectangles A = 9(11) = 99 cm 2 A = 7.8 x 9 2 A = 99 x 3 = 297 cm 2 A = 35.1 cm 2 click to reveal x 2 70.2 cm 2 click to reveal 89

Surface Area 9 cm Answer 9 cm 7.8 cm 11 cm 9 cm Triangles A = 7.8 x 9 2 Rectangles A = 35.1 cm 2 x 2 A = 9(11) = 99 cm 2 70.2 cm 2 click to reveal A = 99 x 3 = 297 cm 2 click to reveal 90

40 Find the surface area of the shape below. 1. Draw and label ALL faces; use the net if it's helpful 2. Find the correct dimensions for each face Answer 3. Calculate the AREA of EACH face 4. Find the SUM of ALL faces 5. Label Answer 47 ft 50 ft 21 ft 42 ft 91

Find the Surface Area. 9 cm 3 cm 4 cm 15 cm 5 cm 6 cm 92

9 cm 3 cm 4 cm 15 cm 5 cm Answer 6 cm Top Rectangle Side Rectangles Bottom Rectangle Trapezoids 12 5 6 12 x 15 x 15 x 15 + 6 180 75 90 18 x 2 x 4 click to reveal click to reveal 150 72 / 2 = 36 x 2 click to reveal 72 click to reveal 93

41 Find the surface area of the shape below. 8 cm 6 cm Answer 9 cm 10 cm 94

42 Find the surface area of the shape below. 2 cm 10 cm Answer m c 6 6 c m 10 cm 95

Surface Area of Pyramids Return to Table of Contents 96

Surface Area of Pyramids What is a pyramid? Polyhedron with one base and triangular faces that meet at a vertex click to reveal How do you find Surface Area? Sum of the areas of all the surfaces of a 3­D Figure click to reveal 97

Find the Surface Area. 17.5 cm go on to see steps 7 cm 17.4 cm 8 cm 98

Find the Surface Area. Answer Bottom Front/Back Left/Right Rectangle Triangles Triangles 8 A = 1 A = 1 2 bh(2) 2 bh(2) x 7 2 56 cm A = 1 A = 1 2 (8)(17.4)(2) 2 (7)(17.5)(2) A = 139.2 cm 2 A = 122.5 cm 2 99

Surface Area Find the surface area of a square pyramid with base edge of 4 inches and triangle height of 3 inches. 3 in Base 4 Triangles Surface Area 4 16 x 4 + 24 16 40 in 2 click to reveal click to reveal click to reveal 4 in 100