3. The meaning of area: why we need formulas

• This unit is about area. What does that word mean?

The word is never used by itself. It's always followed by the word of and then by the name of a thing. Here are some examples.

• The area of a piece of paper 8.5 inches wide and 11 inches high is 93.5 square inches.
• The area of a rectangle is its base times its height.
• The area of the United States is about 3.5 million square miles.

• In math language, a piece of paper is called a region in the plane. Draw the plane as a big piece of graph paper with square boxes that measure one inch by one inch. The area of each box is one square inch. Each box is called a unit square.

Very often, we don't mention the measurement: we might say that the unit square's length and width are each 1 unit, and the unit square's area is one square unit.

And sometimes we say just the number and don't mention the word 'unit' either. It's OK to say that a rectangle has base 5, height 3, and area 15.

The definition of area

• Two regions are called congruent if they have exactly the same size and shape. In other words, either region can be moved in the plane, or flipped over, so that it exactly covers the other.
Now we can completely describe in practical terms what area means:

• Congruent regions have the same area.
• The area of a square with length 1 and width 1 is 1.
• Suppose you cut a region into pieces. To find the area of the region, add the areas of the pieces.

• One way to try to figure out the area of a region is to cut it into unit squares. For example, a rectangle with base 3 and height 2 can be cut into 6 unit squares, and so its area is 1 + 1 + 1 + 1 + 1 + 1 = 6. So you might want to say: the area of a region is the number of unit squares it contains.

• Unfortunately, most regions don't contain an exact number of unit squares. To see this, we can drag some regions onto a piece of graph paper. There will usually be a messy strip around the edge of the region that doesn't get cut up into unit squares.

•To deal with the problem, you could try using smaller squares, but you run into a serious problem: To get within one percent of the right answer, you need to use ten thousand squares!

There is a better way: In the next lessons, we will figure out simple rules, called formulas, to find the area of regions that you know about.

Instructions

• In the figure at the right, click on the triangle near its center and drag it onto the grid. If you click near the right angle, you will rotate the triangle, whose exact area is 200 square units Then click 'Find Area.' Do this several times. Each time, the picture will show

• red squares that are completely inside the region. These must be counted as 1 unit in the area.
• pink squares completely outside the region. These are not part of the area.
• white border squares that lie partly inside and partly outside the triangle . It's not clear whether to count these in the area. To estimate the area, the best you can do is to count each of these as 1/2 unit.
The 'results' pane shows the area estimate, the percentage error for this try, and the average percentage error. Can you get the error to be less than one per cent? Try dragging the triangle into a few different positions. You can see that the method of counting squares doesn't work all that well.