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Surface Area and Volume(9.2.1)

Surface Area and Volume(9.2.1). Surface Area of Prisms. Surface Area = The total area of the surface of a three-dimensional object (Or think of it as the amount of paper you’ll need to wrap the shape.) Prism = A solid object that has two identical ends and all flat sides.

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Surface Area and Volume(9.2.1)

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  1. Surface Area and Volume(9.2.1)

  2. Surface Area of Prisms Surface Area = The total area of the surface of a three-dimensional object (Or think of it as the amount of paper you’ll need to wrap the shape.) Prism =A solid object that has two identical ends and all flat sides. We will start with 2 prisms – a rectangular prism and a triangular prism.

  3.  https://www.youtube.com/watch?v=ZJ-VMcbLTaU Triangular Prism Rectangular Prism Base Base

  4. Surface Area (SA) of a Rectangular Prism Like dice, there are six sides (or 3 pairs of sides)

  5. Prism net - unfolded

  6. Add the area of all 6 sides to find the Surface Area. 6 - height 5 - width 10 - length

  7. SA = 2lw + 2lh + 2wh SA = 2 (10 x 5) + 2 (10 x 6) + 2 (5 x 6) = 2 (50) + 2(60) + 2(30) = 100 + 120 + 60 = 280 units squared SA = 2lw + 2lh + 2wh 6 - height 5 - width 10 - length OR use this Formula for finding the Surface Area of any Prism SA = B x 2 + H(perimeter of the base) SA = (5)(6) x 2 + 10(5 + 6 + 5 + 6) SA = 30 x 2 + 10(26) =280 units squared

  8. Practice SA = 2lw + 2lh + 2wh = 2(22 x 10) + 2(22 x 12) + 2(10 x 12) = 2(220) + 2(264) + 2(120) = 440 + 528 + 240 = 1208 ft squared 12 ft 10 ft 22 ft OR SA = area of base(2) + H(perimeter of base) SA = (10)(12)(2) + (22)(10+12+10+12) SA = 1208 ft squared

  9. SA = area of base(2) + H(perimeter of base) SA = 5 x 6 x 2 + 12(5 + 6 +5 + 6) SA = 60 + 12(22) OR SA = 60 + 264 SA = 2lw + 2lh + 2wh SA = 2(12)(6)+2(5)(6) + 2(12)(5) SA = 144 + 60+ 120 SA = 324 u²

  10. SA = area of base(2) + H(perimeter of base) SA = 6.5 x 3 x 2 + 7(6.5 + 6.5 +3 + 3) SA = 39 + 7(19) OR SA = 39 + 133 SA = 2lw + 2lh + 2wh SA = 2(6.5)(3)+2(3)(7) + 2(6.5)(7) SA = 39 + 42+ 91 SA = 172 u²

  11. SA = area of base(2) + H(perimeter of base) SA = 4.5 x 3.2 x 2 + 4.2(4.5 + 4.5 +3.2 + 3.2) SA = 28.8 + 4.2(15.4) OR SA = 28.8 + 64.68 SA = 2lw + 2lh + 2wh SA = 2(4.5)(3.2)+2(4.2)(3.2) + 2(4.5)(4.2) SA = 28.8 + 26.88+ 37.8 SA = 93.48 u²

  12. Surface Area of a Triangular Prism • 2 bases (triangular) • 3 sides (rectangular) Base SA = B x 2 + H(perimeter of the base)

  13. Unfolded net of a triangular prism

  14. 2(area of triangle) + Area of rectangles Area Triangles = (b x h) 2 = (12 x 15) 2 = (180) 2 = 90 Area Rect. 1 = b x h = 12 x 25 = 300 Area Rect. 2 = 25 x 20 = 500 15ft SA = 90 + 90 + 300 + 500 + 500 SA = 1480 ft squared

  15. 9 cm Practice 2 Triangles = (b x h) 2 = (8 x 7) 2 = (56) 2 = 2(28) cm Rectangle 1 = 10 x 8 = 80 cm 2 Rectangles = 2(9 x 10) = 2(90) cm Add them all up SA = 28 + 28 + 80 + 90 + 90 SA =316cm2 7 cm 8 cm 10 cm 2 x B + H(perimeter of the base) 2 bh + 10 ( 8 + 9 + 9) 2 2 (8)(7) + 10(26) 2 OR 112 + 10(26) 2 316 cm² 56 + 260 =

  16. 2 x B + H(perimeter of the base) 2 Triangles = (b x h) 2 = (3 x 4) 2 = (12) 2 = (6)2 cm Rectangle 1 = 5 x 7 = 35 cm Rectangle1 = 4 x 7 = 28 cm Rectangle1 = 3 x 7 = 21 cm Add them all up SA = 6 + 6 + 35+ 28+ 21 SA = 96 cm2 bh x 2 + 7( 5 + 4 + 3) 2 OR (3)(4) x2 + 7(12) 2 12 x 2+ 7(12) 2 96 cm² 12 + 84 =

  17. 2 x B + H(perimeter of the base) 2 Triangles = (b x h) 2 = (8 x 3) 2 = (24) 2 = (12)2 cm 3 Rectangles = 8 x 12 = 96 cm Add them all up SA = 12 + 12 + 96+ 96 + 96 SA = 312 cm2 bh x 2 + 12( 8 + 8 + 8) 2 (8)(3) x2 + 12(24) 2 OR 24 x 2+ 12(24) 2 312 cm² (2)12 + 288 =

  18. Surface Area of a CylinderReview • Surface area is like the amount of paper you’ll need to wrap the shape. • You have to “take apart” the shape and figure the area of the parts. • Then add them together for the Surface Area (SA)

  19. Parts of a cylinder A cylinder has 2 main parts. A rectangle and A circle – well, 2 circles really. Put together they make a cylinder.

  20. The Soup Can Think of the Cylinder as a soup can. You have the top and bottom lid (circles) and you have the label (a rectangle – wrapped around the can). The lids and the label are related. The circumference of the lid is the same as the length of the label.

  21. Area of the Circles Formula for Area of Circle A=  r2 = 3.14 x 32 = 3.14 x 9 = 28.26 But there are 2 of them so….. 28.26 x 2 = 56.52 units squared

  22. The Rectangle This has 2 steps. To find the area we need base and height. Height is given (6) times the base which is the circumference. The base of the rectangle is the same as the distance around the circle (or the Circumference).

  23. Find Circumference Formula is C =  x d = 3.14 x 6 (radius doubled) = 18.84 (base) The Circumference is your base then multiply it times the height of the can. A = b x h = 18.84 x 6 (the height given) = 113.04 units squared

  24. Add them together Now add the area of the circles (remember there are two!) and the area of the rectangle (circumference) together. 56.52 + 113.04 = 169.56 units sq. The total Surface Area!

  25. Formula SA = 2 ( r2) + H( d) 2 x Lids Label Area of Circles Area of Rectangle

  26. PracticeBe sure you know the difference between a radius and a diameter! SA = 2 ( r2) + H( d) = 2(3.14 x 112) + 14(3.14 x 22) = 2 (3.14 x 121) + (967.12) = 2(379.94) + (967.12) = (759.88) + (967.12) = 1727 cm2

  27. More Practice! SA = 2 ( r2) + ( d x H) = 2 ( 3.14 x 5.52) + (3.14 x 11 x 7) = 2 (3.14 x 30.25) + (241.78) = 2 (94.985) + (241.78) = (189.97) + (241.78) = 431.75 cm2

  28. SA = 2 ( r2) + ( d x H) = 2 (3.14 x 42) + (3.14 x 8 x 12) = 2 (50.24) + (301.44) = (100.48) + (301.44) = 401.92 cm2

  29. SA = 2 ( r2) + ( d x H) = 2 (3.14 x 7.52) + (3.14 x 15 x 9.4) = 2 (176.625) + (442.74) = (353.25) + (442.74) = 795.99 cm2

  30. Lesson 9.2.3Volume of prisms and Cylinders • The number of cubic units needed to fill the shape.Find the volume of this prism by counting how many cubes tall, long, and wide the prism is and then multiplying. • There are 24 cubes in the prism, so the volume is 24 cubic units. 4 – length 2 x 3 x 4 = 243 3 – width 2 – height

  31. Formula for Prisms

  32. The volume V of a prism is the area of its base B times its height H. V = BH Try It V = BH Find area of the base = (8 x 4) x 3 = (32) x 3 Multiply it by the height = 96 ft3 3 ft - height 4 ft - width 8 ft - length

  33. Practice V = BH = (22 x 10) x 12 = (220) x 12 = 2640 cm3 12 cm 10 cm 22 cm

  34. V = BH V = (5 x 6) x 12 V = (30) x 12 V = 360 u3

  35. V = BH V = (5 x 4.5) x 8 2 V = (10.625) x 12 V = 127.5 cm3

  36. V = BH V = (16 x 12) x 10 2 V = (96) x 10 V = 960 cm3

  37. V = BH V = (9 x 12) x 18 2 V = (54) x 18 V = 972 cm3

  38. V = BH V = (18 x 2) x 3 2 V = (18) x 3 V = 54 ft.3

  39. Cylinders

  40. Try It V = r2H The radius of the cylinder is 5 m, and the height is 4.2 m Substitute the values you know. V = 3.14m · 52m · 4.2m V =329.7m3

  41. Practice 13 cm - radius 7 cm - height V = r Start with the formula V = 3.14 x 132 x 7 substitute what you know = 3.14 x 169 x 7 Solve using order of Ops. = 3714.62 cm3

  42. V = r2H V = 3.14 x 32 x 10 V = 3.14 x 9 x 10 V = 282.6 in3

  43. V = r2H V = 3.14 x 62 x 18 V = 3.14 x 36 x 18 V = 2,034.72 in3

  44. V = r2H V = 3.14 x 82 x 21 V = 3.14 x 64 x 21 V = 4,220.16 u3

  45. V = r2H V = 3.14 x 5.52 x 16 V = 3.14 x 30.25 x 16 V = 1,519.76 in3

  46. Lesson Quiz Find the volume of each solid to the nearest tenth. Use 3.14 for . 1. 2. 4,069.4 m3 861.8 cm3 3. triangular prism: Base area = 24 ft2, Height = 13 ft 312 ft3

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