Right Circular Cylinders A right circular cylinder is like a right prism except that its bases are congruent circles instead of congruent polygons.
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1 Volume-Lateral Area-Total Area page #10 Right Circular Cylinders A right circular cylinder is like a right prism except that its bases are congruent circles instead of congruent polygons. base height base radius Lateral Area of a Right Circular Cylinder: Imagine the diagram shows a can with its ends removed. If you cut along AB, you can unroll the metal and lay it out flat. B A h B A Circumference of the base C = π d h The lateral area becomes the area of this rectangle: L.A. = πdh or L.A. = 2 πrh Total Area of a Right Circular Cylinder: The total area of a right circular cylinder is the lateral area plus the area of the two bases: T.A. = πdh + 2 πr 2 or T.A. = 2 πrh + 2 πr 2 Volume of a Right Circular Cylinder: The volume of a right circular cylinder is the area of one base times the height. V = Bh V = πr 2 h
2 Volume-Lateral Area-Total Area page #11 Example 1: Find the lateral area, total area and volume of the following right circular cylinder: L.A. = 2 πrh = 2 π(3)(4) = 24 π cm 2 3 cm 4 cm T.A. = 2 πrh + 2 πr 2 T.A. = 24 π + 2 π(3) 2 = 24 π + 18 π = 42 π cm 2 V = πr 2 h = π(3) 2 (4) = 36 π cm 3 Example 2: Find the radius of the cylinder if its height is 10 meters and its volume is 90 π square meters. V = Bh V = πr 2 h 90 π = πr 2 (10) 90π 10π = 10πr2 10π 9 = r 2 3 meters = r Problems: 1. Find the lateral area, total area and volume of the following right circular cylinder: 3 2
3 Volume-Lateral Area-Total Area page #12 2. Find the radius of a cylinder if its height is 6 feet and its volume is 36 π feet Find the height of a cylinder if its radius is 4 feet and its volume is 48 π ft The total area of a cylinder is 40 π. If h = 8, find r.
4 Volume-Lateral Area-Total Area page #13 Right Circular Cone In the right circular cone, label the base, radius, height of the cone and slant height. Lateral Area, Total Area and Volume of a Right Circular Cone Lateral Area: πr Total Area: Lateral Area + B Total area: πr + πr 2 h V = 1 3 Bh r V = 1 3 πr2 h
5 Volume-Lateral Area-Total Area page #14 Example 1: Find the lateral area, total area and volume of the cone: 8 10 r =10 2 r =100 r 2 = 36 r = 6 r Lateral Area: π(6)(10) = 60 π Total Area: 60 π + π(6) 2 = 60 π + 36 π = 96 π Volume: 1 3 π(6)2 (8) = 1 3 π(36)(8) = 1 π(288) = 96 π 3 Example 2: The volume of a right circular cone is 48 π cubic units and its altitude is 4 units. Find the radius of the base. V = 1 3 πr2 h 48π = 1 3 πr 2 (4) 144π = πr 2 (4) 144 = 4r 2 36 = r 2 6 = r
6 Volume-Lateral Area-Total Area page #15 Problems: 1. Find the lateral area, total area and volume of the cone r 2. Find the height of a right cone whose volume is 924 π cm 3, and whose base has a radius of 14 cm. 3. Find the radius of a cone with lateral area of 72 π and slant height of 9.
7 Volume-Lateral Area-Total Area page #16 4. The radius of the base of a cone measures 3 units and the altitude measures 7 units. Find the volume. 5. The volume of a cone is 320 π cm 3 and the altitude measures 15 cm. Find the length of the radius of the base. 6. Water is pouring into a conical reservoir at the rate of 1.8 m 3 per minute. How long will it take to fill the reservoir? 5.2 m 6.8 m
8 Volume-Lateral Area-Total Area page #17 S p h e r e s Area = 4πr 2 r Volume = 4 3 πr3 Example 1: The diameter of a sphere is 8. Find the area and volume. Since d = 8, r = 4. Area = 4 πr 2 Volume = 4 3 πr3 Area = 4 π(4) 2 Volume = 4 3 π(4)3 Area = 4 π(16) Volume = 4 3 π(64) Area = 64 π Volume = 256π 3 Example 2: The area of a sphere is 256 π. Find the volume. 4 πr 2 = 256 π r 2 = 64 r = 8 V = 4 3 πr3 V = 4 3 π(8)3 V = 4 3 π(512) V = 2048π 3
9 Problems: Volume-Lateral Area-Total Area page #18 1. Find the area and volume of the sphere Find the area and volume of the sphere A baseball has a radius 7 cm long. Find its surface area. 4. Find the volume of a sphere with a 3 cm radius.
10 Volume-Lateral Area-Total Area page #19 5. Find the length of a radius of a sphere whose area is 196 π cm How many cm 3 of air can be pumped into a basketball if its maximum diameter is 20 cm? 7. Find the length of a radius of a sphere whose volume is 972 π cm The area of a sphere is 400 π. Find the volume.
11 Volume-Lateral Area-Total Area page #20 Formulas B = the area of one base 1. Right Prism L.A. = sum of the areas of the lateral faces T.A. = L.A. + 2B h V = Bh h 2. Right Circular Cylinder L.A. = 2 πrh r h T.A. = L.A. + 2B = 2 πrh + 2 πr 2 V = Bh = πr 2 h 3. Regular Pyramid L.A. = sum of the areas of the lateral faces h T.A. = L.A. + B V = 1 3 Bh 4. Right Circular Cone L.A. = πr h r T.A. = L.A. + B = πr + πr 2 V = 1 3 Bh = 1 3 πr2 h 5. Sphere r A = 4πr 2 V = 4 3 πr3
12 Volume-Lateral Area-Total Area page #21 Review Volume, Lateral Area, Total Area 1. Find the lateral area, total area and volume: 15 m 10 m 12 m 9 m 2. The walls and ceiling of a warehouse are to be painted. How many square meters must be covered if the warehouses is 120 meters by 96 meters with a 3 meter high ceiling? 3. Find the lateral area, total area and volume:
13 Volume-Lateral Area-Total Area page #22 4. A side of the base of a square pyramid measures 6 units and a lateral edge measures 5 units. Find the total area. 5. Find the lateral area, total area, and volume: Find the radius of a right circular cylinder with lateral area of 216 π and height of 12.
14 7. Find the lateral area, total area and volume: Volume-Lateral Area-Total Area page # r 8. What is the slant height of a cone with lateral area of 9π 10 and radius of 1 5? 9. Find the area and volume: 4
15 Volume-Lateral Area-Total Area page # The area of a sphere is 484 π. Find the volume of the sphere. 11. A pyramid has a rectangular base 10 cm long and 6 cm wide. The pyramid s height is 4 cm. Find the volume. 12. A cone has a radius 8 and height 6. Find the lateral area, total area and volume.
16 13. The total area of a cylinder is 18 π. If h = 8, find r. Volume-Lateral Area-Total Area page # The volume of a cylinder is 72 π. If h = 8, find L.A. 15. The area of the base of a cone is 49 π square units and the slant height measures 20 units. Find the length of the altitude.
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