Geometry Notes Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: • Calculate the volume of given geometric figures. • Calculate the surface area of given geometric figures. • Solve word problems involving volume and surface area. Vocabulary: As you read, you should be looking for the following vocabulary words and their definitions: • volume • surface area • sphere • great circle of a sphere • pyramid • cone Formulas: You should be looking for the following formulas as you read: • volume of a rectangle solid • surface area of a rectangular solid • volume of a cylinder • surface area of a cylinder • volume of a solid with a matching base and top • volume of a sphere • surface area of a sphere • volume of a pyramid • volume of a cone We will continue our study of geometry by studying three-dimensional figures. We will look at the two-dimensional aspect of the outside covering of the figure and also look at the three-dimensional space that the figure encompasses. The surface area of a figure is defined as the sum of the areas of the exposed sides of an object. A good way to think about this would be as the surface area
Geometry Notes Volume and Surface Area Page 2 of 19 area of the paper that it would take to cover the outside of an object without any overlap. In most of our examples, the exposed sides of our objects will polygons whose areas we learned how to find in the previous section. When we talk about the surface area of a sphere, we will need a completely new formula. The volume of an object is the amount of three-dimensional space an object takes up. It can be thought of as the number of cubes that are one unit by one unit by one unit that it takes to fill up an object. Hopefully this idea of cubes will help you remember that the units for volume are cubic units. Surface Area of a Rectangular Solid (Box) SA = 2(lw + lh + wh ) l = length of the base of the solid w = width of the base of the solid h = height of the solid Volume of a Solid with a Matching Base and Top V = Ah A= area of the base of the solid h = height of the solid Volume of a Rectangular Solid (specific type of solid with matching base and top) V = lwh l = length of the base of the solid w = width of the base of the solid h = height of the solid volume
Geometry Notes Volume and Surface Area Page 3 of 19 Example 1: Find the volume and the surface area of the figure below 2.7 m 4.2 m 3.8 m Solution: This figure is a box (officially called a rectangular prism). We are given the lengths of each of the length, width, and height of the box, thus we only need to plug into the formula. Based on the way our box is sitting, we can say that the length of the base is 4.2 m; the width of the base is 3.8 m; and the height of the solid is 2.7 m. Thus we can quickly find the volume of the box to be V = lwh = ( 4.2)(3.8)(2.7) = 43.092 cubic meters. Although there is a formula that we can use to find the surface area of this box, you should notice that each of the six faces (outside surfaces) of the box is a rectangle. Thus, the surface area is the sum of the areas of each of these surfaces, and each of these areas is fairly straight-forward to calculate. We will use the formula in the problem. It will give us SA = 2(lw + lh + wh ) = 2( 4.2 * 3.8 + 4.2 * 2.7 + 3.8 * 2.7) = 75.12 square meters. A cylinder is an object with straight sides and circular ends of the same size. The volume of a cylinder can be found in the same way you find the volume of a solid with a matching base and top. The surface area of a cylinder can be easily found when you realize that you have to find the area of the circular base and top and add that to the area of the sides. If you slice the side of the cylinder in a straight line from top to bottom and open it up, you will see that it makes a rectangle. The base of the rectangle is the circumference of the circular base, and the height of the rectangle is the height of the cylinder. cylinder
Geometry Notes Volume and Surface Area Page 4 of 19 Volume of a Cylinder V = Ah A = the area of the base of the cylinder h = the height of the cylinder Surface Area of a Cylinder SA = 2(πr 2 ) + 2πrh r = the radius of the circular base of the cylinder h = the height of the cylinder π = the number that is approximated by 3.141593 Example 2: Find the volume and surface area of the figure below 10 in 12 in Solution: This figure is a cylinder. The diameter of its circular base is 12 inches. This means that the radius of the circular base is 1 1 r = d = (12) = 6 inches. The height of the cylinderi s 10 inches. 2 2 To calculate the volume and surface area, we simply need to plug into the formulas. Surface Area: SA = 2(πr 2 ) + 2πrh = 2(π ⋅ 62 ) + 2π (6)(10) = 72π + 120π = 192π square units. This is an exact answer. An approximate answer is 603.18579 square units.