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Exploring Cylinders and Prisms: Areas and Volumes

Understand how to calculate the lateral and surface areas, as well as volumes of cylinders and prisms through practical examples and formulas. Apply concepts of bases, lateral faces, and edges to solve geometry problems efficiently.

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Exploring Cylinders and Prisms: Areas and Volumes

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

  2. 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

  3. 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

  4. Cylinders – Surface Area & Volume Cylinder 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)

  5. 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 = 282.74 V= Bh = V = πr² h  need to find r and h V= π(r)²h = 9π(12) = 108π = 339.29

  6. 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 = 452.39 V= Bh = V = πr² h  need to find r and h V= π(r)²h = 16π(14) = 224π = 703.72

  7. 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

  8. 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) = 256 + 128 = 384 square units V = B l = w h l V = (8)(8)(8) = 512 cubic units

  9. 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) = 80 + 120 + 48 = 248 square units V = B l = w h l V = (4)(6)(10) = 240 cubic units

  10. 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 = 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 lwhere h is the height of the triangular base! V = ½ (6)(4)(15) = 180 cubic units

  11. 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.

  12. r 1.8, h 1.8 Example 1-3a 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 Use a calculator. Answer: The volume is approximately 18.3 cubic cm.

  13. Example 3-2c 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 = 48 Simplify LA = Ph P = 12 + 2c c² = 8² + 6² = 64 + 36 = 100 c = 10 Pythagorean Thrm P= 12 + 20 = 32 LA = Ph = 32 (10) = 320 SA = 2B + LA = 2 (48) + 320 = 416 Answer: 416 units2

  14. VBh Volume of a prism 1500 Simplify. Example 1-1a Find the volume of the triangular prism. Answer: The volume of the prism is 1500 cubic centimeters.

  15. 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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