On an improved postulate for introducing the concept of area
by Peter Duveen

PETER'S NEW YORK, Saturday, June 23, 2012--Recently in my teaching, I have found myself re-examining some of the concepts that I had earlier taken for granted. One of these is the simple formula for the area of a rectangle, which is generally expressed as the length times the width. This may seem like a simple idea, but my impression is that it is not at all self-evident. Yet the almost universally accepted postulate used to introduce area in geometry textbooks is that the area of a square is equal to the square of its side. 

It is not a priori clear to me that, given two squares of different dimensions, the ratio of the areas enclosed by the squares would be equal to the ratio of the squares of the respective sides of each figure. It is not a satisfying postulate, in my humble opinion.

Tumbling different ideas in my mind, I arrived at what I believe to be a far more satisfactory postulate, which is simply that, given two rectangles of equal width, but of differing lengths, the ratio of the areas enclosed by each of the two rectangles is equal to the ratio of their respective lengths. To me this is far more self evident than the "side-squared of a square" postulate, and from it may be easily derived the conventional formula for the area of a rectangle, and thus, the area of a square as a special case.

To some, this may seem unimportant, but when one considers that we are contemplating the foundation for the entire edifice upon which our practical work on areas is built, it is difficult to understate its significance. The area of a circle, for example, is generally derived from the area of a triangle, which in turn is derived from the area of a rectangle. Areas such as that of the the cone, the frustrum, and the sphere may also be derived from the area of a rectangle. 

I would like to demonstrate for the reader how simple it is to derive the generally accepted formulae for the areas of a rectangle and a square by employing the the postulate I have introduced above. Let me restate it:
Take two rectangles, of differing lengths but of equal widths, and stand them next to each other. It is immediately apparent that the the ratios of the areas enclosed are as the ratios of the lengths of the rectangles. (figure 1)

Using this as our postulate, let us draw a more complex figure, a smaller rectangle drawn in the left portion of a larger rectangle (figure 2).

Let A, B and C represent the areas of the respective rectangles in which they lie. Let L1 (the length of segment ae) and L2 (the length of segment ab) represent the respective lengths of the smaller and the larger rectangle (aegh and abcd) as indicated, and W1 (the length of segment ah) and W2 (the length of segment ad) represent their respective widths.

We first note that, by the area postulate laid down, (A + B) / A = L2 / L1             [equation 1]

By the same postulate, we can say the following:

(A + B + C) / (A + B) = W2 / W1.                                                                    [equation 2]

Multiplying the left and right sides of the first equation by the left and right sides of the second equation respectively yields the following, after simplification:

(A + B + C) / A = (W2 x L2) / (W1 x L1)                                                           [equation 3]

Since (A + B + C) is none other than the area of the larger rectangle, we can conclude that the ratio of the area of the second rectangle to the first is equal to the ratio of the product of the sides of the second rectangle to the product of the sides of the first rectangle. We now have the means of comparing areas of any rectangles if we know the dimensions of their sides.  The same area postulate can be applied to different arrangements of rectangles.

If we reorganize equation 3 a bit, we have

(B/A) x L1 x W1 = L2 x W2                                                                           [equation 4]

In equation 4, we see that the magnitude of one area may be expressed in terms of the magnitude of another, by multiplying the other by a dimensionless constant, B/A. If we consider L1 x W1 to be what is called the unit area, to which all areas are compared for measurement, then it becomes apparent that all other areas may be expressed as our unit area multiplied by a dimensionless positive real number.

Naturally, the area of a square follows as a special case in which the sides of the rectangle are of equal length.

What has been gained? We have derived our basic area formulae using a self-evident postulate more fundamental than that of assuming that the area of a square is equal to the square of its side.