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# Unit 2 3D Geometry

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### Unit 2 3D Geometry

1. 1. 3-D GEOMETRY Essential Math 30S
2. 2. 3-D OUTCOMES
3. 3. 3-D OUTCOMES
4. 4. 3-D OUTCOMES
5. 5. History of Measurement
6. 6. History of Measurement•Span•Digit•Hand•Cubit•Fathom
7. 7. History of MeasurementClassroom Activity (Page 196)Measure the items listed on page 196using original body parts that underlaythe measurement units listed. Recordthe measurements made by eachperson on the board and compare them.
8. 8. History of MeasurementStudent Class Door Book Desk
9. 9. History of Measurement Notebook Assignment Page 198 – 200 Q. 1 - 7
10. 10. Units of Measure
11. 11. Units of Measure
12. 12. Units of MeasureMetric & Imperial Rulers •The following diagram shows a small ruler with the common metric (centimetres) and imperial (inches) units of linear measurement.
13. 13. Units of MeasureMetric & Imperial Rulers
14. 14. Units of MeasureMetric & Imperial Rulers
15. 15. Units of MeasureCommon Metric and Imperial Units of Length
16. 16. Units of MeasureNotebook AssignmentPage 205 – 208 Q. 1 - 8
17. 17. Unit ConversionConversions Within Systems
18. 18. Unit Conversion
19. 19. Unit Conversion
20. 20. Unit ConversionMetric •10 mm = 1 cm •100 cm = 1 m •1000 m = 1 kmImperial •12 inches = 1 foot •36 inches or 3 feet = 1 yard •5280 feet or 1760 yards = 1 mile
21. 21. Unit ConversionUnit Conversion Ratio To convert a measurement given in one unit of measure to another unit of measure, a unit conversion ratio can be used. A unit conversion ratio is a fraction equal to 1.
22. 22. Unit ConversionUnit Conversion Ratio Examples of unit conversion ratios taken from the table on the previous slide are: The conversion factor can be written with either value in the numerator or the denominator. For example:
23. 23. Unit ConversionUnit Conversion Ratio When converting between units of measure, it is best to write the conversion factor as follows: • The numerator of the ratio consists of the required unit of measure (the unit to which you want to convert). • The denominator of the ratio consists of the given unit of measure (the original units in which the measurement was taken).
24. 24. Unit ConversionExample 1 You purchase 485 cm of wire, however it is sold by the meter. How many meters of wire must you purchase?
25. 25. Unit ConversionExample 2 A plank measures 6 ft, 4 in. How many inches long is the plank?
26. 26. Unit ConversionExample 3 A living room has a length of 5 yards, 2 feet. What is the length of the room in inches?
27. 27. Unit ConversionExample 4 Perform the following calculation: 2 km – 820 m = _______
28. 28. Unit ConversionNotebook Assignment Page 213 - 214 Q. 1 - 8
29. 29. Unit ConversionConversions Between Systems
30. 30. Unit Conversion
31. 31. Unit Conversion
32. 32. Unit ConversionExample 1 Convert 70 miles per hour into km/h.
33. 33. Unit ConversionExample 2 Your dining room measures 5 m x 8 m. How many square yards is this?
34. 34. Unit ConversionNotebook Assignment Page 218 Q. 1 - 5
35. 35. Surface Area• What does it mean to you? • Surface area is found by finding the area of all the sides and then adding those answers up.• How will the answer be labeled? • Units2 because it is area!
36. 36. DefinitionSurface Area – is the total number of unit squares used tocover a 3-D surface.
37. 37. Find the SA of a Rectangular Solid A rectangular solid has 6 faces. Top They are: Top Bottom Front Right Back Side Front Right Side Left Side We can only see 3 faces at any one time. Which of the 6 sides are the same? Top and Bottom Front and Back Right Side and Left Side
38. 38. Surface Area of a Rectangular Solid We know that Each face is a rectangle. Top and the Formula for finding the area of a Right rectangle is: Side A = lw Front Steps: Find: Area of Top Area of Front Area of Right Side Find the sum of the areas Multiply the sum by 2.The answer you get is the surface area of the rectangular solid.
39. 39. Find the Surface Area of the following:Find the Area of each face: 12 m 2 Top 5m A = 12 m x 5 m = 60 m Top Right Side8m Front Front 8m A = 12 m x 8 m = 96 m 2 5m 12 m 12 m 2 2 2 2Sum = 60 m + 96 m + 40 m = 196 m Right 2 Side 8m A = 8 m x 5 m = 40 m 2 2Multiply sum by 2 = 196 m x 2 = 392 m 5m 2 The surface area = 392 m
40. 40. Find the Surface Area Area of Top = 6 cm x 4 cm = 24 2 cm 2 24 m 2 Area of Front = 14 cm x 6 cm = 84 cm 2 Area of Right Side = 14 cm x 4 cm = 56 cm 2 2 56 m Find the sum of the areas:14 cm 84 m 2 2 2 2 24 cm + 84 cm + 56 cm = 164 cm Multiply the sum by 2: 4 cm 2 2 6 cm 164 cm x 2 = 328 cm The surface area of this 2 rectangular solid is 328 cm .
41. 41. NetsA net is all the surfaces of a rectangular solid laid out flat. Back 8 cm Top Left Side Top Right Side 5 cm Right 8 cm Side Front 8 cm Front 8 cm 5 cm 10 cm 5 cm Bottom 10 cm
42. 42. Find the Surface Area using nets. Top Back 8 cm Right Side Front 8 cm Left Side Top Right Side 5 cm 5 cm 8 cm 10 cm Front 8 cmEach surface is a rectangle. 80 A = lw 80 5 cm Find the area of each surface. Bottom Which surfaces are the same? 10 cm 40 Find the Total Surface Area. 50 50 40 What is the Surface Area of the Rectangular solid? 2 340 cm
43. 43. VOLUME AND CAPACITY Essential Math 30S
44. 44. Problem of the DayHow can you cut the rectangular prisminto 8 pieces of equal volume by makingonly 3 straight cuts?
45. 45. Problem of the DayHow can you cut the rectangular prisminto 8 pieces of equal volume by makingonly 3 straight cuts?
46. 46. What is volume and capacity?Volume is the quantity of three-dimensional spaceenclosed by some closed boundary, for example, the spacethat a substance or shape occupies or contains.The volume of a container is generally understood to bethe capacity of the container, that is the amount of fluidthat the container could hold.
47. 47. Warm UpIdentify the figure described.1. two triangular faces and the other faces in theshape of a parallelograms2. one hexagonal base and the other faces in theshape of triangles3. one circular face and a curved lateral surfacethat forms a vertex
48. 48. Warm UpIdentify the figure described.1. two triangular faces and the other faces in theshape of a parallelograms triangular prism2. one hexagonal base and the other faces in theshape of triangles hexagonal pyramid3. one circular face and a curved lateral surface thatforms a vertex cone
49. 49. Volume: Prisms, Cylinders, Pyramids, and Cones
50. 50. Volume of a Prism h V Bh Volume Base Area Height Base Area Remember the “Base Area” formula will be determinedby the base shape.
51. 51. Example #1: Finding the Volume of a Prism Find the volume of the regular rectangular prism. V BhB (4)(12) V (48)(12) 3 V 576 ft
52. 52. Volume of a CylinderBase Area Radius V Bh Volume Base Area Height h 2 B r Base Radius Area
53. 53. Example #2 Finding the Volume of a CylinderFind the volume of a cylinder with height 10 cm 5 cmand radius 5cm. V Bh 10 cm 2 B r 2 B (5) 2 B 25 cm V 25 (10) 3 V 250 cm
54. 54. Volume of a Pyramid 1 V BhB Volume 3 Base Area Height
55. 55. Example #3 Finding Volume of a Pyramid Find the volume of a square pyramid with base edges 5 m and height 3 m. 1 3m V Bh 3 B s2 B 52 2 5m B 25 m 1 V (25)(3) 3 3 V 25 m
56. 56. Volume of a Cone 1 V Bh Volume 3 Height Base Area r B
57. 57. Example#4 Finding the Volume of a ConeThe radius of the base of a cone is 6 m. Its height is 13 m.Find the volume. 1 V Bh 3 13 m B r2 2 B 6 B 36 m 2 6m 1 V (36 )(13) 3 3 V 156 m
58. 58. Practice Assignment Worksheet 4.1 Practice Questions 1-7, Pages 90-91
59. 59. A sphere is a 3D figure that iscircular in shape, e.g., a ball. Allpoints on the sphere are equidistantfrom a single point inside the spherecalled the centre.
60. 60. Volume of a SphereSUMMARIZING:Volume (cylinder) = (Area Base) (height)Volume (cone) = Volume (cylinder) /3 = 3Volume (cone) = (Area Base) (height)/3AND 2(Volume (cone)) = Volume (sphere) 2X =
61. 61. Volume of a Sphere 2(Volume (cone)) = Volume (sphere) 2X = 2(Area of Base ) (height) /3= Volume (sphere) 2( r2 )(h)/3= Volume (sphere) r h BUT h = 2r r 2( r2)(2r)/3 = Volume(sphere) 4( r3)/3 = Volume(sphere)
62. 62. Volume of a Sphere 3 4 r  Volume sphere 3 4 r 3 3
63. 63. Find the volume and surface area ofa sphere with radius 12 cm.Volume = (4/3)πr3