Measurement: Understanding what area is
Informally, area is the amount of two-dimensional space that is inside a closed two-dimensional figure.
Understanding what area is measuring
How can we arrange these three pictures by size?
If we compare the pictures two at a time we can see that the small circle fits entirely inside the square, and the square fits entirely inside the larger circle, so this is the correct arrangement by size, largest to smallest:
But sometimes it's harder to decide. Which of these two rectangles is bigger?
That depends on what we mean by "bigger." One is taller; the other is fatter or wider. "Taller" and "wider" are both one-dimensional descriptions, measurements that are taken in one direction only. But the question might mean something like "Which contains more stuff?" In this case, we can't easily tell just by looking. We need a way to measure—that is, to assign a number to—"the amount of stuff" inside a figure.
The area of a geometric shape is the amount of "stuff" the figure is made of, the amount of two-dimensional space that is "inside the figure."
Geometric shapes, themselves, are abstractions -- shape without actual "stuff" -- because any physical object, even hair or paper or the ink on the paper, has some thickness, and so is really three-dimensional. But for students trying to understand what area is, or how to compare the area of one figure to the area of another, thinking about which "has more stuff" can be helpful.
The "amount of stuff" doesn't change if we rotate the figure, or rearrange its parts. So, in defining area, we include these two conditions:
- If two shapes match exactly (are congruent), they contain the same amount of "stuff," so we say they have the same area;
- If we "cut" a shape and rearrange the parts so that no "stuff" was gained or lost—we don't lose any parts, or overlap the "stuff" when we tape the parts back together—the resulting shape has the same area as the original.
Just as we use a standard-sized chunk of length -- like a millimeter, or an inch, or a mile, or a light-year -- to measure other lengths, we will need to use a chunk of area to measure other areas. We could choose a standard of any size and shape, and call its area 1. So, for example, if we decide that this chunk is the standard, then we would call its area "1" and would use it to measure other shapes by comparing it to them. This shape clearly has an area of "2" because it can be built from two pieces we call "1," like this . Similarly, the area of is 3, because we can build that shape from three of the units: .
If we had chosen as the standard, then its area would be called "1" and would have an area of 1/2 (because it uses half the "stuff").
But circles and, in fact, most shapes are not built of whole copies of glued along the edges. No matter what "chunk" of area we might choose for our standard, we will need mathematics, not just copying and pasting, to help us compute other areas.
Defining a standard unit of area: Any shape chunk would do, but by convention, we use a square as the unit. Whatever ruler we use to measure the side of the square also has units. If the side of the square is "one unit" (1 inch, 1 mile, 1 centimeter, 1 yard), then the area of the square is called "one square unit" (one square inch, one square mile...).
Discovering formulas for area
Students who understand area as it is described above can reinvent for themselves most of the area formulas that they are often asked merely to memorize. Each formula they derive helps strengthen their understanding (and memory) for the other formulas they know. (See article on area formulas.)
length, distance, volume, surface area, perimeter, circumference.