Wikijunior:What can you use math for?/Area

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6 {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king 6
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3 {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king {{{square}}} black king 3
2 {{{square}}} white pawn {{{square}}} white pawn {{{square}}} white pawn {{{square}}} white pawn {{{square}}} white pawn {{{square}}} white pawn {{{square}}} white pawn {{{square}}} white pawn 2
1 {{{square}}} white rook {{{square}}} white knight {{{square}}} white bishop {{{square}}} white queen {{{square}}} white king {{{square}}} white bishop {{{square}}} white knight {{{square}}} white rook 1
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Let's say that you, for some reason, wanted to know exactly how many squares there are on a chess board. How would you figure it out? The first thing you think of would probably be to count them out: 1, 2, 3, 4, 5, and so on. You're a busy person though and don't have that kind of time, you need to know quickly. How do you do it? You could use a very simple geometric property: area.

Area is how much space there is inside a 2d object (a 2d [2 dimensional] object is something like a circle or a square but not a ball or box). Area is measured in square units. The units are whatever I am using to measure the object, they can be feet, inches, or miles in the US or meters, centimeters, or kilometers in the rest of the world. In the case of our chess board, the units we will use will be squares.

How do we find the area? For a square (like our chess board) that is simple; all we need to do is multiply the length of the bottom times the length of a side. So here is the process we will follow to find the area of our chess board. First, count the number of squares along the bottom of the board, then count the number of squares along a side. Both numbers should be 8. Next multiply the two numbers. 8 x 8 = 64. So the area of the chess board is 64 square squares, thus we have 64 squares on one chessboard, and we found that out without wasting time counting all of the spaces.

This idea can be applied to any flat, rectangular surface- even if it doesn't have squares marked off. Simply measure a surface's length and width and multiply them. They do not need to even be whole numbers (they can have decimals or fractions). For example, if a table surface measures 150 cm by 250 cm, then its area is 150 x 250 = 37,500. That's a big number! But if you drew 1 cm by 1cm squares on the table, that's how many there would be.

It's hard to think about area with decimals, but it is very useful because not all tables are going to have a whole number of squares. For example, if our table earlier was actually a little bit off- say, 150.2 by 249.6, we can still multiply these numbers, but you might need a calculator. 150.2 x 249.6 = 37,489.92, which is slightly less than before. Notice that even though one side was slightly bigger, the other side was smaller by more, so there is less area. If you drew squares on this table, you would not even get 37,489 full squares, because all squares on the edge would get cut off. But if you added up all of those pieces of squares, 37,4892 is how many full squares could be made (with .92 left over).

Area with decimals has some results that you might not expect. If a square is 1/2 meters by 1/2 meters (.5 m by .5 m), its area is .5 x .5 = .25 or 1/4. That seems strange because it is less than either the length or the width. But if you divide a square by drawing one line vertically through the center and another horizontally through the center, you will have four equal sections. That is, each square is 1/4 of the total area even though it is only 1/2 by 1/2 of the total length and width.

Area can be applied to more than just square surfaces. You can find the area on a circular or triangular table too. In fact, the area of any surface, no matter the shape, can be measured. It gets a lot more difficult for some shapes, however, so we won't teach just how, but be aware it can be done. By using area, you can compare objects that aren't of the same shape. For example, you could determine whether a circular table had more or less space than a square table.