Which cylinders are larger around than top to bottom?

A physicist or engineer who uses π (pi) in numerical calculations may need to have access to 5 or 15 decimal place approximations to this special number, but most of us—mathematicians included—don’t need to know more (decimal-wise) than the fact that it’s roughly 3.14. Yet there is an inexplicable nerdy subculture far removed from real mathematics that obsesses with memorizing huge numbers of decimals of π.

In recent years, March 14 has become synonymous with Pi Day, this year focusing on the fact that pi is approximately 3.141592654. Hence, some have argued, March 14, 2015 at about 9:26:54 (AM or PM? And in which time zone?) is a moment worthy of attention. It's all in good fun, and anything that helps to draw attention to mathematics has to be good, right? Since the correct four-decimal approximation is 3.1416, don't be surprised if we see similar hype this time next year (March 14, 2016).

The decimals of pi have their own fascination and mysteries, which some of the world's best minds have wondered about, but it's probably fair to say that most mathematicians are as entertained by the popularity of pi decimals memorization as creative writers are by spelling bees.

A great bar bet

Let’s ask a related thought-provoking question which makes a great bar bet. Assemble several diverse cans, such as those depicted at the top of this post, and ask: For which of the three (idealized) cylinders shown is the distance from top to bottom longer than the distance around?

Despite the foreshortening due to the photograph having been taken from above, it’s clear that the can on the left isn’t a contender. The middle one looks like it just might make the cut, and our experience is that many people will agree with the claim that the most extreme can on the right is about one and a half times as tall as it is around, thus qualifying handsomely.

One surefire but uninstructive way to determine the answer is to resort to actual measurement, but let’s instead explore some more revealing approaches. It just so happens that the width (8.5 inches) of a sheet of A4 paper matches up nicely with the height of the tallest can, as shown in the image above left.

Now imagine rotating the paper 90 degrees and trying to wrap its width around the red can, as shown in the third image. Instead of going around more than once, it actually falls short, leaving an unmistakable gap (image at right).

Those of us who thought the tallest can was a winner must conclude that our visual perception cannot be trusted: we’re just not very good at gauging girth. Reasoning with mathematics can also play a role here, when one realizes what the tallest can contains three tennis balls (image below).

Ignoring the negligible additional space around (and above) the balls when they are in the can, it follows that the can is 3 times as tall as it is wide. Re-enter our friend pi: the circumference of any can is pi times its width. So it all boils down to knowing which is larger, 3 or pi!

Without knowing a single digit of the decimal expansion of pi, we can conclude that the circumference of the red can (and hence the others as well) exceeds its height. To put it another way, all of the cylinders are larger around than they are from top to bottom.

Ironically, many mathematicians do as poorly with this bar bet as members of the general public, even when told what is in the third can. Perhaps the problem isn't underappreciation of pi in its most basic incarnation here as the ratio of circumference to width for circles (or cylinders). Maybe it's an overappreciation of pie (the gastronomical one). I have to wonder if my own dodgy girth-gauging ability springs from years of looking in the mirror and being in denial that I've gained weight because of my sweet tooth.

Still curious about the decimal part of pi? Take a closer look at the paper gap in the third image above. Its curved extent from left to right is pi minus 3 (namely 0.14159...) times the width of the cylinder. In this way you can literally see the decimal part of pi.

A related bar bet involves finding a straight glass that is so tall that its height exceeds its circumference. It's not as easy "as it looks."

Update 3/17/15: Click this link to read about a similar bar bet that Martin Gardner challenged me on.

Try either version on the next pi decimal memorization enthusiast you run into!

All images by Colm Mulcahy.