Is Area of a Square a Circular Argument?

(New Question of the Week)

I love it when students want to know why something has to be the way it is, and are not satisfied just being told to use a formula. Last month, Shunya asked this kind of question, which gave me a chance to refer to our archive and go beyond it.

A proof that isn’t a proof?

Here is his question:

I was introduced to the concept of Area of square and rectangle, through a problem.

The problem was, what is the area of a rectangle with sides 3 ft and 2 ft.

My teacher solved it by dividing it into parts of smaller squares with sides 1 ft, and then he counted the number of squares which was equal to 6 (6 squares). And then he concluded that the area of rectangle is 6 square feet.

I was unable to understand how he was able found out the area of square with side 1 ft, because he had to multiply 1 ft * 1 ft to get 1 square feet, which meant that he used the formula without proving it.

Can you please explain it to me as to how one can prove that the area of square of side 1 unit by 1 unit is 1 square unit without using other geometrical figures.

I did some searching on Wikipedia and I found this:

https://proofwiki.org/wiki/Axiom:Area_Axioms

It says that “The area of a square with a side of length one unit is defined to be one square unit”.

Can you please explain to me why the area of square is side * side and that of rectangle is length * breadth? (without using the area of square of side 1 unit =1 square unit)

He’s uncovered what appears to be circular reasoning: The formula A = LW is justified by counting squares, but to find the area of one of those squares, we have to use that formula A = LW to get 1 square unit. Is it circular?

He also found that the area of a square is just defined, not proved. Is that legal?

The need to start with a definition

We answered essentially the same question 16 years ago, so I could refer to that and other past answers.  I replied,

Hi, Shunya.

First, we can’t really talk about area without defining what unit we are using; so we can’t avoid talking about a unit square: that is the definition of the unit of area! See this page:

Defining a Square Centimeter

Second, for rectangles with integer sides, the formula for area of a rectangle essentially follows from counting squares, as you suggested:

Units and Square Units in Measuring Perimeter and Area

Figuring Out Formulas for Area

Proving area formulas generally (say, for rectangles with irrational sides, or for curved regions) requires more than just counting; the axioms you found are the basis for doing this. They say, essentially, that (a) we define a unit to use; (b) we can add areas of non-overlapping pieces; and (c) we can rearrange parts of a region and still have the same area. Based on those “obvious” assumptions, we can use calculus and other more advanced means to find areas of complex regions.

Let me know if you have more questions about this.

The first link I gave is an answer to essentially the same question; here is what I said in 2002:

Before we can measure anything, we have to define the unit with which we will measure it. In this case, we define something called a square centimeter as the area of a one-centimeter square. That doesn't need proof, since the concept doesn't exist until we make this definition.

Given that definition, we can find the area of any rectangle (with integral sides, to start with) by laying out H rows of W unit squares. We can count them by multiplying, and find that the area is then W*H.

If we then apply this to the unit square, we of course get 1*1 = 1, but this just shows that our calculation is consistent with the definition. It also motivates our notation, where we write a square centimeter as cm^2 (with an exponent) because

    1 cm * 1 cm = (1*1)(cm*cm) = 1 cm^2

The fact that we are multiplying two centimeter measures makes it reasonable to call this area 1 cm^2. But again, that is not the definition of a square centimeter, only a conclusion from the definition.

So the definition of the unit called a square centimeter comes before the formula; we don’t use the formula to find the area of the square, but use the square to say what we mean by “square centimeter”.

The other two links I gave talk about the (informal) development  of the formula from the concept of measuring area in terms of little squares, as Shunya’s teacher did. The third link extends this from rectangles to other basic shapes such as the triangle and parallelogram (as I have discussed in previous posts, referring to the same page). But that approach to area doesn’t work well for defining area of any figure at all. As I explained above, we can’t cut up little squares and put them together to make a circle; curved shapes require something more. The page Shunya found about axioms is the foundation for formal proofs about area, which can cover (pun intended) any reasonable region of the plane.

Alternatives

But a thoughtful student will never be fully satisfied! (That’s good.) Shunya wrote back, asking why we can’t define area using a different unit:

From your first link (“Defining the Square centimeter”) I understood that we need units for area and so we picked “square units”. But why can’t I define the area of square of side 1 unit as 2 square units? If I did so then I get twice the area of what I would have gotten by defining 1 square unit (for 1 unit side square).

Could we decide to call that unit square an area of 2 units, rather than 1? That is, could we make the unit of area half of the square, rather than the whole thing? And what would happen to our formula?

Great question. And my answer will surprise people who think everything about math has to be the way they learned it:

You certainly can do that! You might think of that unit as a right isosceles triangle (half of the unit square), and call the area of the square “2 triangular units” (e.g. “2 triangular centimeters”, in particular). This is, in fact, done in some problems (such as in studying tangrams, whose basic unit is such a triangle). I don’t think you’d want to call it a “square unit”, though. That would imply the wrong thing. It would just be your area unit.

Here is an example where students are asked to measure the area of tangram pieces using the small triangle as a unit of area (though here they assume you already know about square units):

Tangrams

Here is a version of a picture from my third Ask Dr. Math link, modified to use triangular units:

     +-----------------------+
     | \                     |
     +---+                   |
     | \ | \                 |
     +---+---+               |
     | \ | \ | \             |
     +---+---+---+           |
     | \ | \ | \ | \         |
     +---+---+---+---+       |
     | \ | \ | \ | \ | \     |
     +---+---+---+---+---+   |
     | \ | \ | \ | \ | \ | \ |
     +---+---+---+---+---+---+

This triangle, with base 6 units and height 6 units, has an area of 36 “triangular units” (2 for each square unit). So the formula for area of a triangle is \(A = bh\) in this unit, not \(A = \frac{1}{2}bh\) as it is in square units.

Now for a little fun

I continued, referring to an unarchived discussion we have had:

People have sometimes written to us asking if you could define “circular units”, the area of a unit circle being 1; that would be fine, too. It would be easy to work with for circles, but very awkward for rectangles, and hard to explain to children, since you can’t fit circles together to fill a larger circle.

Ultimately, the main reason for choosing the square unit as we do is convenience: it makes the formulas for familiar figures like the rectangle simpler. Who would want to learn that the area of an L by W rectangle is 2LW? Students would be asking why we don’t define the unit of area in terms of a square, to eliminate the unnecessary 2 …

There are often multiple choices for a definition, and we tend to choose whatever makes other things we do as simple as possible. Something like this is done in physics, too: the unit of force in the metric system, for example, is chosen so that the formula becomes \(F = ma\), rather than \(F = kma\), where k would be some special number you’d have to memorize.

Sometimes people can disagree as to what is “simple”. If you haven’t heard of the “controversy” about replacing π (pi) with a new number called τ (tau), take a look! It comes down to whether you’d rather have the 2 in \(C = 2\pi r\), or a 1/2 in \(A = \frac{1}{2}\tau r^2\). Or something like that.

So much for circular arguments.

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