Learn what area and volume are, how they are related, and how they are each calculated.

The last episode 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 protagonist 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.

## How to Calculate Length

In the article about 1D, 2D, and 3D coordinates, 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 remodel. 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, teaspoons, and tablespoons are all volume measurements used to figure out how much of each ingredient goes into a recipe.

## Wrap Up

That’s all we’re going to say about area and volume for now. At this point, you should understand 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 episode to find out!

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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!