On the Circumference of a Circle

It is proved in Euclid's famous textbook that two circles are to one another in circumference as they are in diameter. Certainly this was known to the Pythagoreans two or three hundred years before, and to earlier calculators in Babylonia and Egypt, even if the idea of proving propositions of this sort had not yet been invented.

The meaning of this proposition is that all circles have the same shape, so to speak. One might say this is obvious; a circle is a circle, after all, not a rectangle or anything else. But circles of different sizes are really not so obviously the same shape as all that. For example, one might speak of the "curvature" of a curve such as a circle, and notice that by any reasonable standard (including the rigorous one used by mathematicians) a small circle shows more curvature than a large one. When a circle is truly enormous, like the equator of the earth, its curvature near any particular point is hardly noticeable, while curvature is instantly observed at any point of the ring one puts on a finger. In what sense, then, has the earth's equator the "same shape" as the wedding‑ring?

That this is not such an obvious question was brought home to me the other day when I arrived at my biweekly poker game. The other players are a mixed group: some retired professors (chemistry, art history), some active professors (psychology, mathematics), a bus driver, a restaurateur, a retired retailer. I am the mathematician. It was the bus driver (I'll call him Fred) who greeted me on my arrival, saying loudly, "O.K., here he comes; now we'll get the answer."

He had put a numerical problem to those who had come before me, but was dissatisfied with the answers he seemed to be getting, and was pleased that there would be a real mathematician to ask, someone whose word he was willing to take without question. Here was his problem: He wanted to put up a trellis, and needed to know how much to buy (at the garden supply shop) if he wanted it to be circular and about eight feet across.

It took me a while to get his words sorted out. I thought at first a trellis was a rigid wooden structure standing vertically, as a sort of porous wall, with maybe a horizontal section as a roof. In the form of a circle? One of my colleagues (a psychologist) saw it as simple. You buy an eight foot square piece and cut it into a circle by sort of removing the corners.

No, No, No! That wasn't what Fred meant at all. I quieted everyone down and asked a couple of questions, and found that the "trellis" Fred had in mind was a length of fencing made of flexible wire, and could be bought in rolls, like barbed wire. One edge was finished, the other edge pronged. When you bought a piece you put it up as a fence, the pronged part downward to sink into the ground; and you could bend it to enclose a piece of earth of whatever shape you liked.

So the problem was simple. The length Fred should buy measures the circumference of his circular plot, whose diameter he wanted to be eight feet. In other words, he needed a little over 25 feet, that being eight times π. The ratio of the circumference to the diameter of a circle is the very definition of the number π, whose value is approximately 3.14159, 22/7 being another handy approximation. So, a little over three times as much as the diameter.

While I was giving him the answer (I thought for a moment and said, "a little over 25 feet, let's see..." and he said, "It doesn't have to be exact..." and I said, "well, I think about two inches over 25 feet..." and he said "No, you don't have to worry about inches," anxious to save me labor. I said I'd write down the result for him, and he was grateful, for he thought he might forget. I tried to explain how I got the answer, by multiplying the desired diameter by π, but this conversation went on in the interstices of the poker game, which was in full swing before I could get to the general idea; and I know I didn't really manage to tell him what I had done.

For, a few minutes later, he said, "Well, if I wanted it to be a little less, like seven feet across, I'd buy a foot less of the trellis, wouldn't I?" I said "No!" with such violence that I startled him. The poker game got in the way after that, though I tried to tell him that for each foot less of diameter he wanted he would buy about three feet less of trellis, well, a little more than three feet less... I could see it was not sinking in. I said I would write down the answer for a seven foot plot if he liked, but he said no, thanks, he had the answer for eight, and if he wanted a little less he'd just get a little less trellis, and the exact measurements didn't really matter that much.

Now I had occasion to tell part of this story to a lady of my acquaintance during the intermission of a concert. This lady is no mathematician, but she is a potter, and she knows circles when she sees them, having daily occasion to ponder their properties. So I told her of the bus driver and his problem with the trellis. She knew instantly what sort of trellis he was talking about, by the way, and understood the problem as stated in his words, where I and my poker companions had had to ferret it out.

So then I came to the question: "O.K. To make a circular plot eight feet across, how much trellis is needed?"

The lady grew flustered. "Oh, that's pi, isn't it. How much is pi? Do we need pi arr squared, here, or is it two pi arr? Is it pi that is three and one‑seventh?..." I stopped her.

"Stop," I said. "Never mind about π and some formula you learned in high school. That's not what I'm after. You know what a circle is. Picture one in your mind. Imagine it is eight feet across, and you have to walk around its rim. How far around is it? Just be reasonable; forget mathematics. How far around will it be?"

She said, "Why, about twenty‑five feet, I suppose." I said, "Very good; that's right. Now how did you get that?" She couldn't say. The nearest thing to a calculation she could come up with was this: that it was plainly more than twice as far around, because going back and forth once was already twice as far, and didn't yet enclose any territory. So it must be about three. But she was vague about it. She relied, I believe, mainly on her memory of what a circle looks like, and her tactile sense, perhaps imagining what becomes of a circular string when unwrapped and measured against a diameter. She had forgotten to worry about π by then, and was grateful that I was not drilling her in mathematics.

R.A.Raimi

On the Circumference of a Circle

May 14, 1994