Relating Length, Surface Area, and Volume

　　The Math IC tests not only whether you’ve memorized the formulas for the different geometric solids, but also whether you understand those formulas. The test gauges your understanding by asking you to calculate the lengths, surface areas, and volumes of various solids. The Math IC will ask you about the relationship between these three properties. The Math IC includes two kinds of questions covering these relationships.　　Comparing Dimensions　　The first way the Math IC will test your understanding of the relationship among the basic measurements of geometric solids is by giving you the length, surface area, or volume of different solids and asking you to compare their dimensions. The math needed to answer comparing-dimensions questions isn’t that hard. But in order to do the math, you need to have a good grasp of the formulas for each type of solid and be able to relate those formulas to one another algebraically. For example,

　　The surface area of a sphere is the same as the volume of a cylinder. What is the ratio of the radius of the sphere to the radius of the cylinder?

　　This question tells you that the surface area of a sphere and the volume a cylinder are equal. A sphere’s surface area is 4π(rs)2, where rs is the radius of the sphere. 　　A cylinder’s volume is π(rc)2　h, where rc is the radius of the cylinder, and h is its height. Therefore,

　　The question asks for the ratio between the radii of the sphere and the cylinder. This ratio is given by r s/rc. Now you can solve the equation 4πrs2 = πrc2　h for the ratio rs/rc.

　　Changing Measurements　　The second way the Math IC will test your understanding of the relationships among length, surface area, and volume is by changing one of these measurements by a given factor, and then asking how this change will influence the other measurements.　　When the lengths of a solid in the question are increased by a single constant factor, a simple rule can help you find the answer:If a solid’s length is multiplied by a given factor, then the solid’s surface area is multiplied by the square of that factor, and its volume is multiplied by the cube of that factor.　　Remember that this rule holds true only if all of a solid’s dimensions increase in length by a given factor. So for a cube or a sphere, the rule holds true when just a side or the radius changes, but for a rectangular solid, cylinder, or other solid, all of the length dimensions must change by the same factor. If the dimensions of the object do not increase by a constant factor—for instance, if the height of a cylinder doubles but the radius of the base triples—you will have to go back to the equation for the dimension you are trying to determine and calculate by hand.　　Example 1

　　If you double the length of the side of a square, by how much do you increase the area of that square?

　　If you understand the formula for the area of a square, this question is simple. The formula for the area of a square is A = s2, where s is the length of a side. Replace s with 2s, and you see that the area of a square quadruples when the length of its sides double: (2s)2 = 4s2.　　Example 2

　　If a sphere’s radius is halved, by what factor does its volume decrease?

　　The radius of the sphere is multiplied by a factor of 1 2 (or divided by a factor of 2), and so its volume multiplies by the cube of that factor: (1 2)3 = 1 8. Therefore, the volume of the sphere is multiplied by a factor of 1 8 (divided by 8), which is the same thing as decreasing by a factor of 8.　　Example 3

　　A rectangular solid has dimensions x　y　z (these are its length, width, and height), and a volume of 64. What is the volume of a rectangular solid of dimensions x /2　y /2　z?

　　If this rectangular solid had dimensions that were all one-half as large as the dimensions of the solid whose volume is 64, then its volume would be (1 2)3　64 = 1 8　64 = 8. But dimension z is not multiplied by 1 2 like x and y. To answer a question like this one, you should use the volume formula for rectangular solids: Volume = l　w　h. It is given in the question that xyz = 64. So, x 2　y 2　z = 1 4　xyz = 1 4　64 = 16.