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Surface Area of Prisms and Cylinders 10-4 Holt Geometry

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Warm Up Find the perimeter and area of each polygon. 1. a rectangle with base 14 cm and height 9 cm 2. a right triangle with 9 cm and 12 cm legs 3. an equilateral triangle with side length 6 cm

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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.

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**Vocabulary lateral face lateral edge right prism oblique prism**

altitude surface area lateral surface axis of a cylinder right cylinder oblique cylinder

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**Prisms and cylinders have 2 congruent parallel bases.**

A lateral face is not a base. The edges of the base are called base edges. A lateral edge is not an edge of a base. The lateral faces of a right prism are all rectangles. An oblique prism has at least one nonrectangular lateral face.

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An altitude of 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 area is 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.

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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.

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**The surface area of a right rectangular prism with length ℓ, width w, and height h can be written as**

S = 2ℓw + 2wh + 2ℓh.

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**The surface area formula is only true for right prisms**

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. Caution!

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**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.

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**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.

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The lateral surface of a cylinder is the curved surface that connects the two bases. The axis of a cylinder is the segment with endpoints at the centers of the bases. The axis of a right cylinder is perpendicular to its bases. The axis of an oblique cylinder is not perpendicular to its bases. The altitude of a right cylinder is the same length as the axis.

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**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 = 2rh = 2(8)(10) = 160 in2 S = L + 2r2 = 160 + 2(8)2 = 288 in2

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**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. C = 2r Circumference of a circle 24 = 2r Substitute 24 for C. r = 12 Divide both sides by 2.

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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 = 2rh = 2(12)(6) = 144 cm2 Lateral area S = L + 2r2 = 144 + 2(12)2 = 432 in2 Surface area

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**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.

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**Example 3: Finding Surface Areas of Composite Three-Dimensional Figures**

Find the surface area of the composite figure.

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

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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 = – 2(8) = 72 cm2

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Check It Out! Example 3 Find the surface area of the composite figure. Round to the nearest tenth.

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**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.

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**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 = — 2()(22) = cm2

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**Always round at the last step of the problem**

Always round at the last step of the problem. Use the value of given by the key on your calculator. Remember!

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**Example 4: Exploring Effects of Changing Dimensions**

The edge length of the cube is tripled. Describe the effect on the surface area.

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**Example 4 Continued original dimensions: edge length tripled: S = 6ℓ2**

24 cm original dimensions: edge length tripled: S = 6ℓ2 S = 6ℓ2 = 6(8)2 = 384 cm2 = 6(24)2 = 3456 cm2 Notice than 3456 = 9(384). If the length, width, and height are tripled, the surface area is multiplied by 32, or 9.

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

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**Check It Out! Example 4 Continued**

11 cm 7 cm 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 Notice than 550 = 4(137.5). If the dimensions are halved, the surface area is multiplied by

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Lesson Quiz: Part I Find the lateral area and the surface area of each figure. Round to the nearest tenth, if necessary. 1. a cube with edge length 10 cm 2. a regular hexagonal prism with height 15 in. and base edge length 8 in. 3. a right cylinder with base area 144 cm2 and a height that is the radius

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Lesson Quiz: Part II 4. A cube has edge length 12 cm. If the edge length of the cube is doubled, what happens to the surface area? 5. Find the surface area of the composite figure.

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