- 1. K.V Faridkot Harkamalpreet Singh Brar 9 -B th
- 2. Topic
- 3. Objectives At the end of the lesson the students should be able; To find the surface area of a cylinder ..
- 4. What is a cylinder? The term Cylinder refers to a right circular cylinder. Like a right prism, its altitude is perpendicular to the bases and has an endpoint in each base.
- 5. PRESENTATION base altitude radius base
- 6. What will happen if we removed the end of the cylinder and unrolled the body? Lets find out !!!!
- 7. This will happen if we unrolled and removed the end of a cylinder…. h Circumference of the base 2Πr 2
- 8. Notice that we had formed 2 circles and a 1 rectangle…. The 2 circles serves as our bases of our cylinder and the rectangular region represent the body
- 9. How can we solved the surface area of a Cylinder? To solve the surface area of a cylinder, add the areas of the circular bases and the area of the rectangular region which is the body of the cylinder.
- 10. This is the formula in order to solved the surface are of a cylinder. SA= area of 2 circular bases + are of a rectangle oR
- 11. We derived at this formula..!! SA=2Πr2 +2Πr Or SA=2Πr (r + h)
- 12. Find the surface area of a cylindrical water tank given the height of 20m and the radius of 5m? (Use π as 3.14) Given: SA=2πr2 +2πrh h=20m r=5m =2(3.14)(5m)2 + 2[(3.14) (5m)(20m) =157m2 + 628m SA =785m2
- 13. 2-Surface Area of a Prism Cubes and Cuboids
- 14. Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. A cuboid has 6 faces. The top and the bottom of the cuboid have the same area.
- 15. Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. A cuboid has 6 faces. The front and the back of the cuboid have the same area.
- 16. Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. A cuboid has 6 faces. The left hand side and the right hand side of the cuboid have the same area.
- 17. Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. Can you work out the 5 cm 8 cm surface area of this cubiod? The area of the top = 8 × 5 = 40 cm2 7 cm The area of the front = 7 × 5 = 35 cm2 The area of the side = 7 × 8 = 56 cm2
- 18. Surface area of a cuboid To find the surface area of a shape, we calculate the total area of all of the faces. 5 cm So the total surface area = 8 cm 2 × 40 cm2 Top and bottom 7 cm + 2 × 35 cm2 Front and back + 2 × 56 cm2 Left and right side = 80 + 70 + 112 = 262 cm2
- 19. Formula for the surface area of a cuboid We can find the formula for the surface area of a cuboid as follows. Surface area of a cuboid = w l 2 × lw Top and bottom h + 2 × hw Front and back + 2 × lh Left and right side = 2lw + 2hw + 2lh
- 20. Surface area of a cube How can we find the surface area of a cube of length x? All six faces of a cube have the same area. The area of each face is x × x = x2 Therefore, x Surface area of a cube = 6x2
- 21. Checkered cuboid problem This cuboid is made from alternate purple and green centimetre cubes. What is its surface area? Surface area =2×3×4+2×3×5+2×4×5 = 24 + 30 + 40 = 94 cm2 How much of the surface area is green? 48 cm2
- 22. Surface area of a prism What is the surface area of this L-shaped prism? 3 cm To find the surface area of 3 cm this shape we need to add together the area of the two 4 cm L-shapes and the area of the 6 rectangles that make up 6 cm the surface of the shape. Total surface area = 2 × 22 + 18 + 9 + 12 + 6 + 6 + 15 5 cm = 110 cm2
- 23. Using nets to find surface area It can be helpful to use the net of a 3-D shape to calculate its surface area. Here is the net of a 3 cm by 5 cm by 6 cm cubiod. 6 cm Write down the area of each 3 cm 18 cm2 3 cm 6 cm face. Then add the 5 cm 15 cm2 30 cm2 15 cm2 30 cm2 areas together to find the surface area. 3 cm 18 cm2 3 cm Surface Area = 126 cm2
- 24. Using nets to find surface area Here is the net of a regular tetrahedron. What is its surface area? Area of each face = ½bh = ½ × 6 × 5.2 = 15.6 cm2 5.2 cm Surface area = 4 × 15.6 = 62.4 cm2 6 cm
- 25. 3-Warm up: Finding the Area of a Lateral Face Architecture. The lateral faces of the Pyramid Arena in Memphis, Tennessee, are covered with steal panels. Use the diagram of the arena to find the area of each lateral face of this regular pyramid.
- 28. Surface Area of a Cone Unit 6, Day 4 Ms. Reed With slides from www.cohs.com/.../229_9.3%20Surface%20Area%20of %20Pyramids%20and%20Cones%20C...
- 29. A cone has a circular base and a vertex that is not in the same plane as a base. In a right cone, the height meets the base at its center. The vertex is directly Height above the center of the circle. Lateral Surface Slant Height r Base r The height of a cone is the perpendicular distance between the vertex and the base. The slant height of a cone is the distance between the vertex and a point on the base edge.
- 30. Surface Area of a Cone Surface Area = area of base + area of sector = area of base + π(radius of base)(slant height) S = B + π r l = π r + π rl 2 l B =πr 2 r
- 31. Lateral Area of a Cone Since Lateral Area = Surface Area – area of the base = π r + π rl L.A. = 2
- 32. Example 1: Find the surface area of the cone to the nearest whole number. a. 4 in. r = 4 slant height = 6 S = π r + π rl 2 = π (4) + π (4)(6) 2 6 in. = 16π + 24π = 40π = 40(3.14) ≈ 126in. 2
- 33. Example 2: Find the surface area of the cone to the nearest whole number. b. l 5 ft. 12 ft. First, find the slant height. Next, r = 12, l = 13. l =r +h 2 2 2 S = π r + π rl 2 = (12) + (5) 2 2 = π (12) + π (12)(13) 2 = 144 + 25 = 169 = 144π + 156π = 300π l = 169 = 13 ≈ 942 ft. 2
- 34. On your own #1 Calculate the surface area of: S = π r 2 + π rl •S = π(7)2 + π(7)(11.40) •S = 49π + 79.80π •S = 128.8π
- 35. On your own #2 Calculate the lateral area of: S = π r = π rl L.A. + 2 •L.A. = π(5)(13) •L.A. = 65π
- 36. Homework Work Packet: Surface Area of Cones
- 37. 4-Surface Area of a Sphere
- 38. Sphere
- 39. Hemisphere
- 40. Great Circle
- 43. (Surface Area of a Sphere) = 4πr2
- 44. 5-Basic Geometric Properties Volume of a cuboid
- 45. In this lesson you will learn to calculate the volume of a cuboid
- 46. Cuboids
- 47. Look at this cuboid Now imagine it is full of cubic centimetres 6 cm 1 cm3 4 cm 10 cm Can you see that there are 10 × 4 = 40 cubic centimetres on the bottom layer? There are 6 layers of 40 cubes making 40 × 6 = 240 cm3
- 48. Let us go back and look at what we did here 6 cm height 4 cm breadth 10 cm length When we worked out the volume we multiplied the length by the breadth and then by the height Volume of a cuboid = length × breadth × height or V=lbh
- 49. Lets us look again at the same cuboid and this 6 cm time try the formula 4 cm 10 cm V=lbh = 10 × 4 × 6 cm3 = 240 cm3 You will see that this is the same answer as we got before
- 50. 6-Volume of a Cylinder
- 51. What is Volume? The volume of a three-dimensional figure is the amount of space within it. Measured in Units Cubed (e.g. cm3)
- 52. Volume of a Prism Volume of a Prism is calculated by Volume = Area of cross section x perpendicular height V = Ah V = (4 x 4) x 4 = 64 m3
- 53. What is this? It has 2 equal shapes at the base, but it is not a prism as it has rounded sides It is a Cylinder
- 54. Volume of a Cylinder How might we find the Volume of a Cylinder?
- 55. Example V = Ah
- 56. Pieces Missing Find the volume of concrete used to make this pipe Volume of Concrete = Volume of Big Cylinder – Volume of Small Cylinder (hole)
- 57. What shape is present here?
- 58. What 3D shapes can you see?
- 59. HOME WORK Find the Volume of the Solid. To 1 decimal place
- 61. Volume of a Cylinder How might we find the Volume of a Cylinder? V = Ah –=
- 62. Conversion of units 1cm – 10mm 1m – 100cm 1km – 1000m
- 63. Conversions of Units 1 cm2 = 10 mm x 10 mm =100 mm2 1 m2 = 100 cm x 100 cm = 10 000 cm2 1 m2 = 1000 mm x 1000 mm = 1 000 000 mm2 1 ha = 100 m x 100 m = 10 000 m2 1 km2 = 100 ha
- 64. What about when cubic units? 1 cm3 = 1cm x 1cm x 1cm = 10 mm × 10 mm × 10 mm = 1000 mm3 1 m3 = 1m x 1m x 1m = 100 cm × 100 cm × 100 cm = 1 000 000 cm3
- 65. Capacity Volume - The volume of a three-dimensional figure is the amount of space within it. Measured in Units Cubed (e.g. cm3) Volume and capacity are related. Capacity is the amount of material (usually liquid) that a container can hold. Capacity is measured in millilitres, litres and kilolitres.
- 67. How does Volume relate to Capacity? 1000 mL = 1 L 1000 L = 1 kL 1 cm3 = 1 mL 1,000cm3 = 1000ml = 1L 1 m3 = 1000 L = 1 kL
- 68. Examples Convert 1800 mL to L 1800ml = 1800/1000 = 1.8L 2.3 m3 to L 1m3 = 1000L (1kL) 2.3m3 = 2.3kL = 2300L
- 69. Length = 5.53cm Capacity Find the Capacity of this cube Length = 5.53cm V = Ah = (5.53 x 5.53) x 5.53 = 169.11cm3 (1cm3 = 1ml) Capacity = 169.11ml
- 70. Example Find the capacity of this rectangular prism. Solution Volume = Ah = (26 x 12) x5 = 312 × 5 = 1560 cm3 (1cm3 = 1mL) Capacity = 1560 mL or 1.56 L (1000mL = 1L)
- 71. Ex 11.08 – Q 7.
- 72. What size rainwater tank would be needed to hold the run-off when 40 mm of rain falls on a roof 12 m long and 3.6 m wide? (Answer in litres.)
- 74. Volume of Cylinders Volume = Base x height V = Bh B Base area = π r2 r h
- 75. Compare Cone and Cylinder Use plastic space figures. Fill cone with water. Pour water into cylinder. Repeat until cylinder is full. r r h
- 76. Volume of Cone? = 3 cones fill the cylinder, so… Volume = ⅓ Base x height
- 77. Volume of Cone 3 cones fill the cylinder Volume = ⅓ Base x height V = ⅓ Bh h = 7 cm Base area = π r2 V = ⅓ (π . 2.5 2) . 7 r =2.5 cm V = ⅓ 3.14 . 6.25 . 7
- 78. 8-Developing the Formula for the Volume of a Sphere
- 79. Volume of a Sphere Using relational solids and pouring material we noted that the volume of a cone is the same as the volume of a hemisphere (with corresponding dimensions) Using “math language” Volume (cone) = ½ Volume (sphere) Therefore 2(Volume (cone)) = Volume (sphere) OR = +
- 80. Volume of a Sphere We already know the formula for the volume of a cone. Volumecylinder Volumecone = 3 OR = ÷3
- 81. Volume of a Sphere AND we know the formula for the volume of a cylinder Volumecylinder = ( Area of Base ) X (Height ) Height BASE
- 82. Volume of a Sphere SUMMARIZING: Volume (cylinder) = (Area Base) (height) Volume (cone) = Volume (cylinder) /3 = ÷3 Volume (cone) = (Area Base) (height)/3 AND 2(Volume (cone)) = Volume (sphere) 2X =
- 83. Volume of a Sphere 2(Volume (cone)) = Volume (sphere) 2X = 2(Area of Base) (height) /3= Volume (sphere) 2( πr2)(h)/3= Volume (sphere) r BUT h = 2r h r 2(πr )(2r)/3 = Volume(sphere) 2 4(πr3)/3 = Volume(sphere)
- 84. 3 4π r Volume of a Sphere 4 π r 3 3 3 3 4π r Volumesphere = 3 4 π r 3 4 π r 3 3 3

- Discuss the meaning of surface area. The important thing to remember is that although surface area is found for three-dimensional shapes, surface area only has two dimensions. It is therefore measured in square units.
- Stress the importance to work systematically when finding the surface area to ensure that no faces have been left out. We can also work out the surface area of a cuboid by drawing its net ( see slide 51 ). This may be easier for some pupils because they would be able to see every face rather than visualizing it.
- Pupils should write this formula down.
- As pupils to use this formula to find the surface area of a cube of side length 5 cm. 6 × 5 2 = 6 × 25 = 150 cm 2 . Repeat for other numbers. As a more challenging question tell pupils that a cube has a surface area of 96 cm 2 . Ask them how we could work out its side length using inverse operations.
- Discuss how to work out the surface area that is green. Ask pupils how we could write the proportion of the surface area that is green as a fraction, as a decimal and as a percentage.
- Discuss ways to find the surface area of this solid. We could use a net of this prism to help find the area of each face.
- Links: S3 3-D shapes – nets S6 Construction and Loci – constructing nets