488 CHAPTER 8 Geometry Cube A cube is a rectangular solid with the same length, width, and height (see the figure to the left). If we call the length of the side of the cube s and use the volume and surface area formulas for a rectangular solid, substituting s in for l, w, and h, we obtain V s s = ⋅ ⋅s s3 = and SA s s s s s s s s s s 2 2 2 2 2 2 6 . 2 2 2 2 =⋅⋅+⋅⋅+⋅⋅= + + = 5 units 3 × units, or 15 square units. The front and back sides of the rectangular solid each has an area of 2 units 3 × units, or 6 square units. The top and bottom sides of the rectangular solid each has an area of 5 units 2 × units, or 10 square units. Therefore, the surface area of the rectangular solid is 2(5 3) 2(2 3) 2(5 2) 2(15) × + × + × = + 2(6) 2(10) 30 12 20 62 + = + + = square units. In general, the surface area of any rectangular solid is SA lw wh lh 2 2 2 . = + + Volume and Surface Area of a Rectangular Solid V lwh SA lw wh lh 2 2 2 = = + + s s s Volume and Surface Area of a Cube V s SA s6 3 2 = = Cylinder Now consider the right circular cylinder shown in Fig. 8.36(a). When we use the term cylinder in this book, we mean a right circular cylinder. The volume of the cylinder is determined by multiplying the area of the circular base, r ,2 π by the height, h, to get V r h. 2 π = The surface area of the cylinder is the sum of the area of the top, the area of the bottom, and the area of the side of the cylinder. Both the top and bottom of the cylinder are circles with an area of r .2 π To determine the area of the side of the cylinder, examine Fig. 8.36(b) and (c). Notice how when flattened, the side of the cylinder is a rectangle whose length is the circumference of the cylinder and whose width is the height of the cylinder. Thus, the side of the cylinder has an area of r h 2 , π ⋅ or rh 2 . π Therefore, the surface area of the cylinder is r r rh 2 , 2 2 π π π + + or SA rh r 2 2 .2 π π = + 2pr (a) (c) h r A = pr2 C = 2pr (b) h h A = 2prh Figure 8.36 Volume and Surface Area of a Cylinder V r h SA rh r 2 2 2 2 π π π = = + Volume of a Cylinder
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