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数学英语 31 What are Area and Volume?

时间:2010-11-19 03:49来源:互联网 提供网友:fi1171   字体: [ ]
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by Jason Marshall

The last article about how to use math to send encrypted messages ended with a bit of a cliffhanger. The good news is that you will soon find out how the story ends. But, as I warned last time, the bad news is that that isn’t going to happen quite yet. Because, in order to escape, the protagonist1 of our story first needs to learn a few things about measuring “sizes” in one, two, and three-dimensional space—which means that today we’re talking about area and volume.
But first, the podcast edition of this article was sponsored by Go to Meeting. With this meeting service, you can hold your meetings over the Internet and give presentations, product demos and training sessions right from your PC. For a free, 45 day trial, visit GoToMeeting.com/podcast.
 
How to Calculate Length
In the article about 1D, 2D, and 3D coordinates2, we learned that a one-dimensional object exists along a line. We also talked about using the number line first mentioned in the article on negative numbers and integers as an example of this one-dimensional object. Now, let’s imagine that the number line is instead a long straight road, and that the various tick marks representing the locations of positive and negative integers are actually mile markers—with the zero marker being located at the initial position of your car. If you were to drive down the road and stop at mile marker ten, you could figure out the number of miles between that marker and mile marker zero (which is the origin where you started) by counting off how many one mile long “unit” measuring sticks you could lay down end-to-end between the two points. Of course, this gives the same result as simply calculating: 10 – 0 = 10.
So, “size” in one dimension is a measure of the length of a line segment. Of course, it’s also a quantity you calculate all the time in the real world using a tape measure: How wide is a room? How tall are you? How far is it to your destination? And numerous other similar things too.
How to Calculate Area
Okay, that’s 1D. What about 2D? Well, first of all, as we talked about in the article on 1D, 2D, and 3D coordinates, a two-dimensional object is something that exists in a plane. As an example of a 2D object, imagine a square with sides that are each 5 meters long. While we’re at it, let’s also imagine another square with sides that are each 1 meter long. This second square is what we’ll call the “unit” square. In the exact same way that we used unit measuring sticks to figure out the length of lines in one-dimension, we can use unit squares to figure out the two-dimensional size—called the area—of our big square. How? Well, the area of a square is just the number of unit squares that fit inside it. So, in the case of our big 5 meter by 5 meter square, 5 unit squares will fit inside of it from left-to-right (since the unit squares are only 1 meter wide), and 5 unit squares will fit across it in the other direction from top-to-bottom. That means a total of 5 x 5 = 25 unit squares will fit inside our big square. Since each unit square has an area of 1 meter x 1 meter = 1 square meter, the area of the big square is 25 square meters. Don’t be confused by the “square meters”—it just indicates that something with a length in meters has been multiplied by something else with a length in meters—and that this is therefore a 2D quantity.
So, “size” in two dimensions is a measure of the area of two-dimensional objects. And this is a very handy quantity to calculate when you’re trying to figure out something like how much carpeting you need to buy to finish your bedroom remodel3. In this case, you need to measure the length and width of the room in feet (or whatever other unit you prefer), and then multiply these two numbers together to find the total area of the room in square feet—that’s how much carpeting you need to buy. Incidentally, it’s also the same number you’d have gotten if you counted up how many 1 foot x 1 foot squares of carpeting you’d need to completely cover your floor.
How to Calculate Volume
And in 3D? Indeed, just as we extended the idea of “size” from 1D to 2D, the same idea holds when moving from 2D to 3D. As we learned in the article on 1D, 2D, and 3D coordinates, a three-dimensional object exists in the three-dimensional space that we live in. As an example of a 3D object, let’s use a cube with sides that are each 5 meters long. And, just as we did with the square in 2D, let’s also imagine another cube with sides that are each one meter long. You guessed it: this second cube is our “unit” cube, and we can use it to figure out the three-dimensional size—called the volume—of our big three-dimensional cube by seeing how many unit cubes will fit into the big cube. Since 5 unit cubes will fit across the big cube from left-to-right, and 5 more unit cubes will fit inside it from top-to-bottom, and yet another 5 unit cubes will fit inside it in the third and final dimension from forward-to-backward—a total of 5 x 5 x 5 = 125 unit cubes will fit inside the big cube. Since the unit cube has a volume of 1 meter x 1 meter x 1 meter = 1 cubic meter, the total volume of the big cube is 125 cubic meters. Again, don’t be confused by the “cubic meters”—it just indicates that we’re talking about a 3D volume.
So, “size” in three dimensions is a measure of the volume of three-dimensional objects. Just like area, volume is a very useful quantity that you use all the time in the real world. When you fill your car with gasoline, you’re putting a certain number of gallons or liters of fuel into it. Gallons and liters are both units of volume—in other words, they’re just different ways of measuring a certain volume of liquid. Volume measurements show up all the time in cooking too—cups, quarts, teaspoons4, and tablespoons are all volume measurements used to figure out how much of each ingredient goes into a recipe.
Wrap Up
Okay, that’s all we’re going to say about area and volume for now. At this point, you should understanding how area and volume are related and how to picture their meaning. In the future, we’ll return to these concepts and talk about how to calculate the area and volume of objects like circles, triangles, and spheres. But, in the meantime, now that we’re armed with a basic knowledge of the meaning of area and volume, we’re ready to find out how our favorite mathematically-inclined secret-agent is faring. Be sure to check out the next article to find out!
Thanks again to our sponsor this week, Go To Meeting. Visit GoToMeeting.com/podcast and sign up for a free 45 day trial of their online conferencing service.
Please email your math questions and comments to.............You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
 


点击收听单词发音收听单词发音  

1 protagonist mBVyN     
n.(思想观念的)倡导者;主角,主人公
参考例句:
  • The protagonist reforms in the end and avoids his proper punishment.戏剧主角最后改过自新并避免了他应受的惩罚。
  • He is the model for the protagonist in the play.剧本中的主人公就是以他为模特儿创作的!
2 coordinates 8387d77faaaa65484f5631d9f9d20bfc     
n.相配之衣物;坐标( coordinate的名词复数 );(颜色协调的)配套服装;[复数]女套服;同等重要的人(或物)v.使协调,使调和( coordinate的第三人称单数 );协调;协同;成为同等
参考例句:
  • The town coordinates on this map are 695037. 该镇在这幅地图上的坐标是695037。 来自《简明英汉词典》
  • The UN Office for the Coordination of Humanitarian Affairs, headed by the Emergency Relief Coordinator, coordinates all UN emergency relief. 联合国人道主义事务协调厅在紧急救济协调员领导下,负责协调联合国的所有紧急救济工作。 来自《简明英汉词典》
3 remodel XVkx1     
v.改造,改型,改变
参考例句:
  • Workmen were hired to remodel and enlarge the farm buildings.雇用了工人来改造和扩建农场建筑。
  • I'll remodel the downstairs bedroom first.我先要装修楼下那间房间。
4 teaspoons 2373c24f8a940bcba4d8d55a1e80e98c     
n.茶匙( teaspoon的名词复数 );一茶匙的量
参考例句:
  • Add two teaspoons of salt. 加两小匙盐。
  • Add 3 heaped teaspoons of sugar. 加满满的三匙糖。 来自《简明英汉词典》
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