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11.2

11.2. Surface Area of Prisms and Cylinders. 11.2 Surface Area of Prisms and Cylinders. Prisms and cylinders have 2 congruent parallel bases.

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11.2

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  1. 11.2 Surface Area of Prisms and Cylinders

  2. 11.2 Surface Area of Prisms and Cylinders Prisms and cylinders have 2 congruent parallel bases. A lateral faceis not a base. The edges of the base are called base edges. A lateral edgeis not an edge of a base. The lateral faces of a right prismare all rectangles. An oblique prismhas at least one nonrectangular lateral face.

  3. 11.2 Surface Area of Prisms and Cylinders An altitudeof a prism or cylinder is a perpendicular segment joining the planes of the bases. The height of a three-dimensional figure is the length of an altitude. Surface areais the total area of all faces and curved surfaces of a three-dimensional figure. The lateral area of a prism is the sum of the areas of the lateral faces.

  4. 11.2 Surface Area of Prisms and Cylinders The net of a right prism can be drawn so that the lateral faces form a rectangle with the same height as the prism. The base of the rectangle is equal to the perimeter of the base of the prism.

  5. 11.2 Surface Area of Prisms and Cylinders The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as S = 2ℓw + 2wh + 2ℓh.

  6. 11.2 Surface Area of Prisms and Cylinders Caution! The surface area formula is only true for right prisms. To find the surface area of an oblique prism, add the areas of the faces.

  7. 11.2 Surface Area of Prisms and Cylinders Example 1A: Finding Lateral Areas and Surface Areas of Prisms Find the lateral area and surface area of the right rectangular prism. Round to the nearest tenth, if necessary. L = Ph P = 2(9) + 2(7) = 32 ft = 32(14) = 448 ft2 S = Ph + 2B = 448 + 2(7)(9) = 574 ft2

  8. 11.2 Surface Area of Prisms and Cylinders Example 1B: Finding Lateral Areas and Surface Areas of Prisms Find the lateral area and surface area of a right regular triangular prism with height 20 cm and base edges of length 10 cm. Round to the nearest tenth, if necessary. L = Ph = 30(20) = 600 ft2 P = 3(10) = 30 cm S = Ph + 2B The base area is

  9. 11.2 Surface Area of Prisms and Cylinders Check It Out! Example 1 Find the lateral area and surface area of a cube with edge length 8 cm. L = Ph = 32(8) = 256 cm2 P = 4(8) = 32 cm S = Ph + 2B = 256 + 2(8)(8) = 384 cm2

  10. 11.2 Surface Area of Prisms and Cylinders The lateral surfaceof a cylinder is the curved surface that connects the two bases. The axis of a cylinderis the segment with endpoints at the centers of the bases. The axis of a right cylinderis perpendicular to its bases. The axis of an oblique cylinderis not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis.

  11. 11.2 Surface Area of Prisms and Cylinders

  12. 11.2 Surface Area of Prisms and Cylinders Example 2A: Finding Lateral Areas and Surface Areas of Right Cylinders Find the lateral area and surface area of the right cylinder. Give your answers in terms of . The radius is half the diameter, or 8 ft. L = 2rh = 2(8)(10) = 160 in2 S = L + 2r2 = 160 + 2(8)2 = 288 in2

  13. 11.2 Surface Area of Prisms and Cylinders Example 2B: Finding Lateral Areas and Surface Areas of Right Cylinders Find the lateral area and surface area of a right cylinder with circumference 24 cm and a height equal to half the radius. Give your answers in terms of . Step 1 Use the circumference to find the radius. Circumference of a circle C = 2r Substitute 24 for C. 24 = 2r Divide both sides by 2. r = 12

  14. 11.2 Surface Area of Prisms and Cylinders Example 2B Continued Find the lateral area and surface area of a right cylinder with circumference 24 cm and a height equal to half the radius. Give your answers in terms of . Step 2 Use the radius to find the lateral area and surface area. The height is half the radius, or 6 cm. L = 2rh = 2(12)(6) = 144 cm2 Lateral area S = L + 2r2 = 144 + 2(12)2 = 432 in2 Surface area

  15. 11.2 Surface Area of Prisms and Cylinders Check It Out! Example 2 Find the lateral area and surface area of a cylinder with a base area of 49and a height that is 2 times the radius. Step 1 Use the circumference to find the radius. A = r2 Area of a circle 49 = r2 Substitute 49 for A. Divide both sides by  and take the square root. r = 7

  16. 11.2 Surface Area of Prisms and Cylinders Check It Out! Example 2 Continued Find the lateral area and surface area of a cylinder with a base area of 49and a height that is 2 times the radius. Step 2 Use the radius to find the lateral area and surface area. The height is twice the radius, or 14 cm. Lateral area L = 2rh = 2(7)(14)=196in2 Surface area S = L + 2r2 = 196 + 2(7)2 =294 in2

  17. 11.2 Surface Area of Prisms and Cylinders Example 3: Finding Surface Areas of Composite Three-Dimensional Figures Find the surface area of the composite figure.

  18. 11.2 Surface Area of Prisms and Cylinders A right triangular prism is added to the rectangular prism. The surface area of the triangular prism is Example 3 Continued The surface area of the rectangular prism is . . Two copies of the rectangular prism base are removed. The area of the base is B = 2(4) = 8 cm2.

  19. 11.2 Surface Area of Prisms and Cylinders Example 3 Continued The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure. S = (rectangular prism surface area) + (triangular prism surface area) – 2(rectangular prism base area) S = 52 + 36 – 2(8) = 72 cm2

  20. 11.2 Surface Area of Prisms and Cylinders Check It Out! Example 3 Find the surface area of the composite figure. Round to the nearest tenth.

  21. 11.2 Surface Area of Prisms and Cylinders Check It Out! Example 3 Continued Find the surface area of the composite figure. Round to the nearest tenth. The surface area of the rectangular prism is S =Ph + 2B = 26(5) + 2(36) = 202 cm2. The surface area of the cylinder is S =Ph + 2B = 2(2)(3) + 2(2)2 = 20 ≈ 62.8 cm2. The surface area of the composite figure is the sum of the areas of all surfaces on the exterior of the figure.

  22. 11.2 Surface Area of Prisms and Cylinders Check It Out! Example 3 Continued Find the surface area of the composite figure. Round to the nearest tenth. S = (rectangular surface area) + (cylinder surface area) – 2(cylinder base area) S = 202 + 62.8 — 2()(22) = 239.7 cm2

  23. 11.2 Surface Area of Prisms and Cylinders Remember! Always round at the last step of the problem. Use the value of  given by the  key on your calculator.

  24. 11.2 Surface Area of Prisms and Cylinders Example 4: Exploring Effects of Changing Dimensions The edge length of the cube is tripled. Describe the effect on the surface area.

  25. 11.2 Surface Area of Prisms and Cylinders 24 cm Example 4 Continued original dimensions: edge length tripled: S = 6ℓ2 S = 6ℓ2 = 6(24)2 = 3456 cm2 = 6(8)2 = 384 cm2 Notice than 3456 = 9(384). If the length, width, and height are tripled, the surface area is multiplied by 32, or 9.

  26. 11.2 Surface Area of Prisms and Cylinders Check It Out! Example 4 The height and diameter of the cylinder are multiplied by . Describe the effect on the surface area.

  27. 11.2 Surface Area of Prisms and Cylinders 11 cm 7 cm Notice than 550 = 4(137.5). If the dimensions are halved, the surface area is multiplied by Check It Out! Example 4 Continued original dimensions: height and diameter halved: S = 2(112) + 2(11)(14) S = 2(5.52) + 2(5.5)(7) = 550 cm2 = 137.5 cm2

  28. 11.2 Surface Area of Prisms and Cylinders Example 5: Recreation Application A sporting goods company sells tents in two styles, shown below. The sides and floor of each tent are made of nylon. Which tent requires less nylon to manufacture?

  29. 11.2 Surface Area of Prisms and Cylinders Example 5 Continued Pup tent: Tunnel tent: The tunnel tent requires less nylon.

  30. 11.2 Surface Area of Prisms and Cylinders Check It Out! Example 5 A piece of ice shaped like a 5 cm by 5 cm by 1 cm rectangular prism has approximately the same volume as the pieces below. Compare the surface areas. Which will melt faster? The 5 cm by 5 cm by 1 cm prism has a surface area of 70 cm2, which is greater than the 2 cm by 3 cm by 4 cm prism and about the same as the half cylinder. It will melt at about the same rate as the half cylinder.

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