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Surface Area Formulas for Prisms and Cylinders

Learn and apply surface area formulas for prisms and cylinders with detailed examples and explanations. Understand key concepts such as base edges, lateral faces, and altitudes. Explore effects of changing dimensions on surface area calculations. 8 Relevant

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Surface Area Formulas for Prisms and Cylinders

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  1. Objectives Learn and apply the formula for the surface area of a prism. Learn and apply the formula for the surface area of a cylinder.

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

  5. 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

  6. 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 cm2 P = 3(10) = 30 cm S = Ph + 2B The base area is

  7. 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

  8. 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.

  9. 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 in. L = 2rh = 2(8)(10) = 160 in2 S = L + 2r2 = 160 + 2(8)2 = 288 in2

  10. 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

  11. 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

  12. Example 3: Finding Surface Areas of Composite Three-Dimensional Figures Find the surface area of the composite figure.

  13. 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.

  14. 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

  15. Example 4: Exploring Effects of Changing Dimensions The edge length of the cube is tripled. Describe the effect on the surface area.

  16. 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.

  17. Check It Out! Example 4 The height and diameter of the cylinder are multiplied by . Describe the effect on the surface area.

  18. 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

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