3-D Shapes Topic 14: Lesson 5 3-D Shapes Topic 14: Lesson 5 Surface Area of Cylinders Holt Geometry Texas ©2007

Objectives and Student Expectations TEKS: G2B, G3B, G6B, G8D, G11D The student will make conjectures about 3-D figures and determine the validity using a variety of approaches. The student will construct and justify statements about geometric figures and their properties. The student will use nets to represent and construct 3-D figures. The student will find surface area and volume of prisms, cylinders, cones, pyramids, spheres, and composite figures. The student will describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed.

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.

Example: 1 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 S = 288 in2

Example: 2 Find the surface area of the composite figure. Round to the nearest tenth.

Example: 2 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.

Example: 2 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 Always round at the last step of the problem. Use the value of  given by the  key on your calculator. Remember!

Example: 3 The height and diameter of the cylinder are multiplied by . Describe the effect on the surface area.

Example: 3 continued original dimensions: height and diameter halved: 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 ¼, or by the scale factor squared.

Example: 4 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?

Example: 4 continued The tunnel tent requires less nylon. Pup tent:

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 with a surface area of 52 cm2 and about the same as the half cylinder with a surface area of approximately 70.8 cm2. It will melt at about the same rate as the half cylinder.