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1 Geometric Solids: Cylinders and Cones

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Cylinders Cylinder: A right prism with circular bases. Therefore, the formulas for prisms can be used for cylinders. The base area is the area of the circle: The lateral area is the area of the rectangle: 2 π rh h 2πr h

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Formulas for Cylinders Surface Area (SA) = 2B + LA = 2 π r ( r + h ) Volume (V) = Bh = h 2πr h Formulas: S.A. = 2 π r ( r + h ) V =

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Example #1 4 For the cylinder shown, find the lateral area, surface area and volume. L.A.= 2 π rh L.A.= 2 π (3)(4) L.A.= 24 π sq. cm. 4 cm 3 cm S.A.= 2 π r 2 + 2 π rh S.A.= 2 π (3) 2 + 2 π (3)(4) S.A.= 18 π +24 π S.A.= 42 π sq. cm. V = π r 2 h V = π (3) 2 (4) V = 36 π

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Cones 5 Cone: A right pyramid with a circular base. Therefore, the formulas for pyramids can be used for cones. The base area of the cone is the area of the circle. The pointy end of the cone is the vertex, or apex. A figure in the shape of a cone is said to be conical. Notice that the height (h) (altitude), the radius and the slant height create a right triangle. l r h

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6 Surface Area ( SA ) = π r ( r + l ) Volume ( V ) = Lateral Area ( LA ) = π r l Where l is the slant height. The base area is the area of the circle: l r h Formulas: S.A. = π r ( r + l ) V = Formulas for Cones

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Example #2: For the cone shown, find the lateral area, surface area, and volume. L.A.= π rl Note: We must use the Pythagorean theorem to find l. L.A.= π (6)(10) L.A.= 60 π sq. cm. 6 cm 8 6 2 +8 2 = l 2 10 S.A.= π r (r + l ) S.A.= π 6 (6 + 10) S.A.= 6 π (16) S.A.= 96 π sq. cm. V= 96 π cubic cm.

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8 Example #3: Find the volume of the cylinder with a radius of 4 inches and a height of 6 inches. Find the volume of the same cylinder with half the radius and the same height. How does the volume change? It is a quarter of the original volume!

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Example #6: 10 rhlL.A.T.A.V 43???? ?26??? rhlL.A.T.A.V 435 26

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Example #7 11 Guess which contains more? *Note, images not perfect OR drawn to scale. Option 1. A can of soda in the shape of a cylinder with a diameter of 5 cm and a height of 10 cm. Option 2. A bottle of soda in the shape of a cylinder AND a cone, with a radius of 4 cm., a height of 10 cm., and a cone height of 6 cm.

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Option 1. The Soda Can

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Option 2. The Bottle ** The bottle is a cylinder with a cone on top.

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The Winner!

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Surface Area and Volume of Cones

Surface Area and Volume of Cones

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