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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**

r – radius h – height Net h h r C Volume (V) = B * h Base Area (B) = π * r2 V = π * r2 * h Surface Area = Lateral Area + Base(s) Area LA = 2π • r • h = circumference * h Bases Area = 2 • π • r2 SA = LA + BA SA = 2π • r • h + 2π • r² = 2πr (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 + 2πr2 need to find r and h SA = 2πrh + 2πr2 = 2π(3)(12) + 2π(3)² = 72π + 18π = 90π = V= Bh = V = πr² h need to find r and h V= π(r)²h = 9π(12) = 108π =

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

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**Prisms – Areas & Volumes**

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

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Example 1 Find the surface area and the volume of the cube to the right 8 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) = = square units V = B l = w h l V = (8)(8)(8) = 512 cubic units

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Example 2 10 4 Find the surface area and the volume of the rectangular prism to the right 6 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) = = square units V = B l = w h l V = (4)(6)(10) = 240 cubic units

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Example 3 Find the surface area and the volume of the isosceles triangle prism to the right c c 4 15 6 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 = and use Pythagorean theorem to find c c² = 3² + 4² c = 5 SA = 2(5)(15) + 6(15) + (6)(4) = = 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

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**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 sq ft. Example 4-2a

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**Find the volume of the cylinder to the nearest tenth.**

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

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**Find the surface area of the triangular prism.**

SA = 2B + LA S.A. of a prism B = ½ b h Base area = area of ∆ B = ½ (12) (8) ∆b=12, ∆h=8 B = Simplify LA = Ph P = c c² = 8² + 6² = = 100 c = 10 Pythagorean Thrm P = = 32 LA = Ph = 32 (10) = 320 SA = 2B + LA = 2 (48) = 416 Answer: 416 units2 Example 3-2c

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**Find the volume of the triangular prism.**

V Bh Volume of a prism 1500 Simplify. Answer: The volume of the prism is 1500 cubic centimeters. Example 1-1a

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**Summary & Homework Summary: Homework:**

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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