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Lesson 12-3, 4, 13-1 Cylinders & Prisms

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Objectives Find lateral areas of cylinders Find surface areas of cylinders Find volume of cylinders Find lateral areas of prisms Find surface areas of prisms Find the volume of prisms

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Vocabulary Axis of a Cylinder – the segment with endpoints that are centers of circular bases Right Cylinder – A cylinder where the axis is also an altitude Oblique Cylinder – a non-right cylinder Bases – congruent faces in parallel planes Lateral faces – rectangular faces that are not bases (not all parallel) Lateral edges – intersection of lateral faces Right Prisms – a prism with lateral edges that are also altitudes Oblique Prisms – a non-right prism Lateral Area – is the sum of the areas of the lateral faces

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Cylinders – Surface Area & Volume Cylinder h r Volume (V) = B * h Base Area (B) = π * r 2 V = π * r 2 * h r – radius h – height Surface Area = Lateral Area + Base(s) Area h C Net LA = 2 π r h = circumference * h Bases Area = 2 π r 2 SA = LA + BA SA = 2 π r h + 2 π r² = 2 π r (r + h)

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Example 1 12 3 Find the surface area and the volume of the cylinder to the right SA = 2πrh + 2πr 2 need to find r and h SA = 2πrh + 2πr 2 = 2π(3)(12) + 2π(3)² = 72π + 18π = 90π = 282.74 V= Bh = V = πr² h need to find r and h V= π(r)²h = 9π(12) = 108π = 339.29

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Example 2 14 8 Find the surface area and the volume of the cylinder to the right SA = 2πrh + 2πr 2 need to find r and h SA = 2πrh + 2πr 2 = 2π(4)(14) + 2π(4)² = 112π + 32π = 144π = 452.39 V= Bh = V = πr² h need to find r and h V= π(r)²h = 16π(14) = 224π = 703.72

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Prisms – Areas & Volumes Regular Triangular Prism Lateral Area (LA) – Sum of each area of the non-base(s) faces of the solid Surface Area (SA) – Sum of each area of (all) the faces of the solid Surface Area = Lateral Area + Base(s) Area h b b b b l LA = 3 b l = Perimeter l Bases Area = 2 ½ b h SA = LA + BA SA = 3 b l + b h base perimeter Rectangular Prism Volume (V) = B h Base Area (B) = L w V = L w h L w h LA = 2 w h + 2 L h Bases Area = 2 L w SA = LA + BA SA = 2(Lw + Lh + wh) Net

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Example 1 Find the surface area and the volume of the cube to the right SA = LA + BA LA = 4· w · l = Perimeter · l and Bases Area = 2 · w h SA = 4 · w · l + 2 w · h h = l = w = 8 SA = 4(8)(8) + 2(8)(8) = 256 + 128 = 384 square units V = B l = w h l V = (8)(8)(8) = 512 cubic units 8

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Example 2 Find the surface area and the volume of the rectangular prism to the right SA = LA + BA LA = 2(w+h) · l = Perimeter · l and Bases Area = 2 · w h SA = 2(w · h) + 2(h · l ) + 2 (w · h) h = 6, l = 10 and w = 4 SA = 2(4)(10) + 2(6)(10) + 2(4)(6) = 80 + 120 + 48 = 248 square units V = B l = w h l V = (4)(6)(10) = 240 cubic units 10 4 6

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Example 3 Find the surface area and the volume of the isosceles triangle prism to the right SA = LA + BA LA = 2 · c · l + 6 · l = Perimeter · l and Bases Area = 2 · (½ b h) SA = 2 · c · l + 6 · l + b · h b = 6, l = 15 and use Pythagorean theorem to find c c² = 3² + 4² c = 5 SA = 2(5)(15) + 6(15) + (6)(4) = 150 + 90 + 24 = 264 square units V = B l = ½ b h l where h is the height of the triangular base! V = ½ (6)(4)(15) = 180 cubic units 15 6 4 c c

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Example 4-2a Find the surface area of the cylinder. The radius of the base and the height of the cylinder are given. Substitute these values in the formula to find the surface area. Surface area of a cylinder Use a calculator. Answer: The surface area is approximately 2814.9 sq ft.

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Example 1-3a Find the volume of the cylinder to the nearest tenth. Answer: The volume is approximately 18.3 cubic cm. The height h is 1.8 cm, and the radius r is 1.8 cm. Use a calculator. Volume of a cylinder r 1.8, h 1.8

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Example 3-2c Find the surface area of the triangular prism. Answer: 416 units 2 S.A. of a prism SA = 2B + LA B = ½ b h Base area = area of ∆ B = ½ (12) (8) ∆b=12, ∆h=8 B = 48 Simplify LA = PhP = 12 + 2c Pythagorean Thrm c² = 8² + 6² = 64 + 36 = 100 c = 10 P = 12 + 20 = 32 LA = Ph = 32 (10) = 320 SA = 2B + LA = 2 (48) + 320 = 416

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Example 1-1a Find the volume of the triangular prism. V BhVolume of a prism 1500Simplify. Answer: The volume of the prism is 1500 cubic centimeters.

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Summary & Homework Summary: –Lateral surface area (LA) is the area of the sides –Base surface area (B) is the area of the top/bottom –Surface area = Lateral Area + Base(s) Area –Prism Volume: V = Bh Surface Area: SA = LA + 2B Triangular and Rectangular prisms on formula sheet –Cylinder Volume: V= πr² h Surface Area: SA = 2πrh + 2πr² = 2πr(r+h) Homework: –pg 692; 7-16

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