Area, Surface Area, and Volume: How to Tell One Formula from Another
Date: 01/18/99 at 22:37:23 From: Sandra Leslies Subject: Surface area,area,volume I am having trouble memorizing the geometric formulas. Say you have the to calculate the volume of a pop can and you have the radius. How can you tell that you are right and that you haven't done area? Thanks.
Date: 01/19/99 at 13:20:43 From: Doctor Peterson Subject: Re: Surface area,area,volume Hi, Sandra. There's a nice simple answer to that last question, though it won't solve everything for you. Suppose you vaguely remember that one formula for a cylinder is pi*r^2*h (pi times radius squared times height), and another is 2*pi*r*h (twice pi times the radius times the height). You can tell which is the area and which is the volume by looking at the dimensions. Suppose the radius is 2 inches and the height is 3 inches, and we accept 3.14 for pi. Then our first formula gives pi*r^2*h = 3.14 * (2 in)^2 * (3 in) = 3.14 * 4 in^2 * 3 in = 37.68 in^3 Do you see how I work with the units just as if they were numbers (or variables in algebra), and end up with the units for the answer? Since the units in^3 are cubic, this is a volume. Similarly, for the second formula 2*pi*r*h = 2 * 3.14 * (2 in) * (3 in) = 6.28 * 6 in^2 = 37.68 in^2 we get square units, in^2, so this is an area. In general, you can just count the dimensions in the formula. r^2*h is the product of 3 dimensions, so it's a 3-dimensional quantity, a volume. r*h is the product of only two dimensions, so it's an area. What this won't do for you is remind you about the constant terms -- is it pi or 2*pi? For a sphere, is it 4*pi*r^2 for area and 4/3*pi*r^3 for volume, or is it the other way around, 4/3*pi*r^3 for area and 4*pi*r^2 for volume? (The first pair is right!) When you get to calculus, you'll learn a trick that helps. I'll show it to you now in case it helps. In calculus, there's a concept called the "derivative." In particular, the derivative of an expression like a * x^n is n * a * x^(n-1) Now, you're multiplying the a by what was the exponent n, and the new exponent of x is decreased by one. You don't need to have any idea what a derivative is. It's easy to figure out, and that's all that matters. In simple cases, the derivative of a volume is the area -- specifically, the area of the surface where the volume is "growing" as you increase the variable for which you took the derivative. Here's an example: The volume of a sphere is 4/3*pi*r^3 The derivative of this is 3*4/3*pi*r^2 = 4*pi*r^2 which is the surface area of a sphere! Here's a harder case: The volume of a cylinder is pi*r^2*h If we take r as the variable that is "growing," the derivative is 2*pi*r^1*h = 2*pi*r*h which is the lateral surface area -- the side of the cylinder, where material would have to be added if you increased the radius. If, instead, we take h as the "growing" variable, the derivative is pi*r^2*1*h^0 = pi*r^2 This is the area of one end of the cylinder, where you have to add material if the cylinder is growing in height. (Unfortunately, this doesn't give you the total area of both top and bottom.) So this method, if you've been able to follow it, can help you check the numbers in a pair of formulas. You still have to memorize them, but at least you can check your memory. Finally, probably the best way to learn these formulas is to know where they come from. The sphere formulas, you may not be ready to figure out on your own; but the cylinder formulas are simple. The lateral surface area is just the circumference of the base circle (2*pi*r) times the height (h) -- picture how you'd make the side of a cylinder by rolling up a rectangle. The volume is the area of the base circle (pi*r^2) times the height (h) -- just like the volume of a rectangular solid. So how can you memorize the formulas? I would suggest you write them all down in a table, and look for relationships. I've just told you how the formulas for a circle and a rectangle combine to give you a cylinder. The more of those you can find, the better. You'll also find some less obvious ones: the volume of a sphere and a cone have an interesting relationship. Make friends with the formulas, and they'll reveal some of their personal secrets to you. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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