8.4 Volume and Surface Area 487 Volume and Surface Area SECTION 8.4 LEARNING GOALS Upon completion of this section, you will be able to: 7 Understand and use formulas to determine the volume and surface area of rectangular solids, cylinders, cones, and spheres. 7 Understand and use formulas to determine the volume of polyhedra, prisms, and pyramids. 7 Solve problems involving cubic unit conversions. Denise is painting her apartment and needs to estimate the amount of paint she will need. The paint can label states, “One gallon will cover about 400 sq. ft.” This sentence refers to the two main geometric topics that we will cover in this section. One gallon refers to volume of the paint in the can, and 400 sq. ft. refers to the surface area that the paint will cover. Why This Is Important Volume and surface area are important geometric concepts that have many real-life applications. These include calculating surface areas to be painted, the amount of sand needed for a volleyball court, the size of a car engine, and the size of an air conditioner needed to cool a home. Understanding volume and surface area is a helpful problem-solving tool for these and many other applications. When discussing a one-dimensional figure such as a line, we can determine its length. When discussing a two-dimensional figure such as a rectangle, we can determine its area and its perimeter. When discussing a three-dimensional figure such as a cube, we can determine its volume and its surface area. Volume is a measure of the capacity of a three-dimensional figure. Surface area is the sum of the areas of the surfaces of a three-dimensional figure. Volume refers to the amount of material that you can put inside a three-dimensional figure, and surface area refers to the total area that is on the outside surface of the figure. Solid geometry is the study of three-dimensional solid figures, also called space figures. Volumes of three-dimensional figures are measured in cubic units such as cubic feet or cubic meters. Surface areas of three-dimensional figures are measured in square units such as square feet or square meters. Rectangular Solids, Cylinders, Cones, and Spheres Rectangular Solid A rectangular solid is a three-dimensional figure in which each surface is a rectangle (see the figure to the left). If the length of a rectangular solid is 5 units, the width is 2 units, and the height is 3 units, the total number of cubes that can be formed is 30 (Fig. 8.35). Thus, the volume is 30 cubic units. The volume of a rectangular solid can also be determined by multiplying its length times width times height; in this case, 5 units 2 units 3 units × × = 30 cubic units. In general, the volume of any rectangular solid is V l w h. = × × The surface area of the rectangular solid in Fig. 8.35 is the sum of the areas of the surfaces of the rectangular solid. Notice that each surface of the rectangular solid is a rectangle. The left and right sides of the rectangular solid each has an area of w h l Research Activities 63. Garfield’s Proof Research the proof of the Pythagorean theorem provided by President James Garfield. Write a brief paper and make a poster of this proof and the associated diagrams. 64. Babylonians and Egyptians The early Babylonians and Egyptians did not have accurate estimations for π and had to devise techniques to approximate the area of a circle. Write a paper on the techniques these societies used to approximate the area of a circle. 65. Heron of Alexandria Write a paper on the contributions of Heron of Alexandria to geometry. 3 units 5 units Left side 2 units Front side Figure 8.35 Ingo Bartussek/Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=