Area and Volume
Date: 04/10/2002 at 18:54:55 From: Dex T. Subject: Volume Dear Dr. Math, I am in seventh grade. I am currently in Pre-Algebra by Saxon Math. I can not figure out how to do volume. Please help me with a simple way to figure it out. Thank you, Dex T.
Date: 04/11/2002 at 14:40:01 From: Doctor Ian Subject: Re: Volume Hi Dex, Do you understand length and area? Or are you a little sketchy on those, too? Length is pretty straightforward, but area can be a little tricky to understand at first. And volume is easiest to understand in comparison to area. A simple way to think about area and volume is this: Two things have the same area if you would need the same amount of paint (or the same amount of material) to cover them up. Two things have the same volume if you would need the same amount of water to fill them up. Note that two things can have different dimensions and still have the same area. An easy way to demonstrate this is to take a sheet of paper, like +-----------+ | | | | | | | | +-----------+ then tear it in half, and put the two halves together like +-----------+-----------+ | | | | | | +-----------+-----------+ Now, we have the same amount of paper, right? So the area has to be the same, even though the two shapes are different. In fact, we could do it again, +-----------+-----------+-----------+-----------+ | | | | | +-----------+-----------+-----------+-----------+ or we could rearrange the pieces to get something that isn't even a rectangle, +-----------+ | | +-----------+-----------+ | | | +-----------+-----------+ | | +-----------+ If you could tear the paper into small enough pieces, you could even rearrange them into something very close to a circle, and it would still have the same area. Volume works the same way, except in three dimensions instead of two. If you go into a store like Lechter's or Williams-Sonoma and look at measuring cups, you can find them in lots of different shapes: round, square, short, tall, fat, thin. But if two measuring cups are marked with the same measurement - say, 1/3 cup - then they have the same volume, because it takes the same amount of stuff to fill them. That is, you can fill one, and if you pour it into the other, it will fill the other one too. So in a sense, area is a way to talk about 'covering', and volume is a way to talk about 'holding'. Now, it turns out that, for lots of shapes, you can _compute_ the area or volume, instead of having to measure it. That's a good thing, because if you're trying to find the area of a building, you don't want to have to paint it. Or if you're trying to find the volume of the building, you don't want to have to fill it with water. Some things are just too big or too small to measure; or they may be inconveniently located (e.g., in orbit around another planet, or at the bottom of the sea); or they might be solid instead of hollow. Basically, there can be lots of reasons why you'd rather compute the area or volume of a thing rather than measuring it. Fortunately, many shapes - like circles, or triangles, or spheres, or pyramids - have formulas that have been worked out. These formulas let you plug in certain information about the shape, and get back the area or the volume. And there are other techniques that can be used to work out the volumes of more complicated shapes. Note that the formulas work in both directions. For example, if I know that each side of a square is 4 inches long, then I can use a formula to work out the area: area(square) = side * side = 4 inches * 4 inches = 16 square inches However, suppose I know the area, and I want to know the length of a side? I can use the same formula to go in the other direction: area(square) = side * side 16 sq in = side * side square root(16 sq in) = square root(side *side) 4 in = side The same sort of thing works for volumes, too. For example, consider a rectangular prism, which is the shape of a cereal box. If you know the length, width, and depth, you can compute the volume of the box: volume(box) = length * width * depth = 8 inches * 4 inches * 2 inches = 64 cubic inches But if you know the volume (perhaps because you measured it), and you know two of the other dimensions, you can get the third: 64 cubic in = length * width * depth = 8 in * 4 in * depth 64 cubic in = 32 square in * depth 64 cubic in / 32 square in = depth = 2 in One of the tricky things about volume is that it's measured in lots of different units. Usually when you _compute_ a volume, you end up with 'cubic units', like cubic inches, cubic centimeters, cubic feet, and so on. However, usually when you _measure_ a volume, you use units like ounces, cups, gallons, liters, and so on. So if you're going to work with volumes a lot, you need to get used to converting back and forth between these kinds of units. But that's a subject for another message. I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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