About Pi

In a Twitter conversation with isologue, the subject of pi (π) came up. It began with:

A certain university student: “Pi? What was it? I can’t remember. Umm, 0, 0, 0.35!!” …Wasn’t pi supposed to have become easier to remember as “3”?? Perhaps we really do need to design institutions treating the entire Japanese population as “the vulnerable”….

In response, I wrote:

“Pi? The area of a circle with radius 1.” That is the sort of answer I would hope for, lol.

Then came the reply:

I cannot even muster the energy to retort, “I said ‘circumference’!”…

So I said:

No, no, that is the correct definition of pi. From that definition, it is easy to prove that the circumference is 2πr, whereas defining it from the circumference makes the proof of πr² somewhat longer, so I think it lacks elegance.

That was the discussion. This is a little too long for Twitter, so I will elaborate here.

First, let us start with the area of an ellipse.

Formula: If an ellipse has horizontal radius a and vertical radius b, its area is abπ.

Proof:

π is the area of a circle with radius 1. Stretch that circle horizontally by a and vertically by b, and its area becomes abπ. QED

Formula: The area of a circle with radius r is πr².

Proof:
From the formula abπ above, rrπ = πr². QED

Formula: The circumference of a circle with radius r is 2πr.

Proof:
Divide the circle into n triangles whose common vertex is the center, and let each triangle have base b and height h.

Next, consider the sum of the lengths of the line segments joining the points that divide the curve, over all such partitions. If that sum has a supremum, define that value as the length of the curve. Then, as n→∞:

nbh/2 → πr², h→r, nb→l (if l exists)

Since l is the circumference:

lr/2 = πr² ⇒ l=2πr

QED

Incredibly simple, isn’t it?^*1 A junior-high-school student—or perhaps even an upper-elementary-school student—could understand this.

By contrast, if π is defined through its relationship to the circumference and one tries to derive πr², one may take as given that the circumference L is the limit of the sum L(n) of the line segments into which the circumference is divided, because that is the definition of a curve’s length. But one cannot take as given that the area of a circle is the limit of the areas of inscribed regular n-gons, so that must first be proved. That is the major difference, I suppose. It is not a very significant difference, but it adds one more step.

If we attempt a similarly naive (that is, rough) proof:

If π is defined as the circumference of a circle with diameter 1, then the circumference of a circle with radius r is 2πr.

Divide the circle into n inscribed isosceles triangles whose common vertex is the center. If their bases are b and their heights h, the sum A of the areas of the n triangles is:

A=nbh/2

As n→∞, by the definition of the length of a curve, nb→2πr and h→r, so:

A=2πrr/2 ⇔ A=πr²

That is what we would like to do. Intuitively, this should be sufficient, but the limit of the inscribed regular n-gons is the inner measure of the circle, not its area, so I suppose we must prove that inner measure = outer measure = area. To do so, we show that the infimum of the areas of circumscribed regular n-gons (the outer measure) is also πr². Then, because inner measure = outer measure = area, the area of a circle with radius 1 is πr².

Reference:

Area of a Circle.

*1 There was an objection: “You don’t know whether l exists! You haven’t defined area either, or shown whether the area of a circle exists!” True….