Ashtray Math

I don’t trust “experts”. I like to work things out for myself. The” experts” might not be experts at all, or they might be wrong, or they might be lying. And anyway, how do you tell if someone is an expert?

Last night I started wondering whether everything I’d learned from “experts” was wrong. And that maybe a lot of the mathematics I use is just plain wrong. There’s a number called π (or Pi) which is the ratio of the circumference of a circle to its radius, or maybe its diameter. And there’s a formula for working out the circumference of a circle using π. And there’s another one for working out the area of a circle using π.

I wondered if I could work all these things out for myself, ignoring untrustworthy mathematics textbooks, written by so-called “experts”.

The first puzzle was how to figure out the value of π. But as I was wondering how to do that, my eyes fell on the cylindrical ashtray in front of me. It was given to me as a present about 40 years ago, and intended to hold water when I was painting watercolours. But, being made of black glazed china, it was no good for that. And so I used it as an ashtray instead. And it’s been my main ashtray for the past 40 years. Amazingly, there’s not a single chip out of it.

I wondered if I could use it to empirically figure out the value of π. It was quite easy to find the approximate diameter of the cylinder with a ruler: it was almost exactly 100 mm. How could I measure its circumference? I got hold of a piece of string, and wrapped it round the ashtray, and marked the points where they overlapped with a felt pen. And then I took the piece of string  and stretched it along the ruler, and found the distance between the marked points was 322 mm. If π = circumference/diameter, then π was 322/100 or 3.22.

I then thought that, since the piece of string I was using was several millimetres thick, that I might get a more accurate measurement using a length of fine cotton, and a biro rather than a felt pen to mark the overlap. This time I measured the circumference of the ashtray as 319 mm. And I had a new value of π as 319/100 or 3.19.

So after about 10 minutes I had a value for π of 3.19. And I could probably get an even more accurate value.

That was step 1. Step 2 was how to figure out the circumference of circle using π. And in fact that is built into the definition of π. For if π = circumference/diameter, then circumference = π × diameter. Or circumference = 2 × π × radius.

Step 3 was to somehow figure out the area of a circle. This looked like a pretty daunting problem. The only things whose areas I know how to measure are rectangles. I know that the area of a room that’s 6 m wide, and 5 m long, is 6 × 5 square metres, or 30 squ m. But what use was that in figuring out the area of a circle?

piI wondered if it could be done using slices of a circle. If there were exactly 12 slices that made up a circle, then if I could find the area of each slice, the area of the circle would be 12 times the slice area. But I couldn’t see how to work out the area of a slice with an arc at one end. After all, I only know how to work out the area of rectangles.

But then I thought that if I used 1000 very, very thin slices, the arc at the end of the slice would get nearer and nearer to a straight line. And would look like this:


So I’d have a triangle with one side of length r, and another side of length c. And this triangle would have exactly half the area of a rectangle of side length c and r. I’d managed to turn a circular problem into a rectangular problem.


The length of c is 1000th of the circumference of the circle, which is 2.π.r. So it must be 2.π.r/1000.

So the area or rectangle r-c is  r × 2.π.r/1000

And the area of the triangular slice is half this, or r × π.r/1000

And the area of the whole circle is 1000 times the area of a slice,  or  r × π.r  (or π.r²).

Like I say, I don’t trust experts. I like to work things out for myself. And today I worked out that the circumference of a circle is 2 × π  × radius, and that the area of a circle is π  × radius × radius, and  that π is roughly equal to 3.19.

And if any “expert” tells me anything different, I won’t believe a word he says, about anything, ever. I’ll only start to respect him a bit if he agrees with me – at least about the circumference and area of circles, and the value of π.

About Frank Davis

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24 Responses to Ashtray Math

  1. Ripper says:

    I’m not an expert, or even very good at maths, but I am an engineer and can see a couple of flaws in your method. It really depends on how accurate you wish to be.

    You can never use pi with complete accuracy, since it has been calculated to a million digits but the popular figure seems to be 3.141, or 3.14. Some people round it up to 3.142. if you had carried on using thinner string and pen you probably would have arrived at the same figure when the string and pen were infinitely thin.

    The second flaw was in squaring the slices. To make a rectangle from triangles you need two right angled triangles, but a slice of a circle, no matter how thin, makes an isoceles triangle, if you ignore the arc on one side. Put two of those together and you get a parrallelogram rather than a rectangle.

    Those two factors, plus the minute arc that would still exist, would combine to throw your calculations off by a considerable amount.

    However, I do not miss the point of the exercise, which was to discover for yourself, so the accuracy doesn’t really matter. But these mathematical principles were discovered by people just like you – true thinkers, in a time when self proclaimed ‘experts’ did not exist.

    • Frank Davis says:

      I believe that pi has an infinite number of digits, and that there have been whole books of pi published, which consist of one long number. I take that to mean that it is impossible to ever know what pi is, and that numbers like 3.14 are just the wrong number.

      And as the number of slices increase, the base angle of the isosceles triangle.comes ever closer to a right angle (much like the ever thinner string and pen). With an infinite number of slices, the angle becomes a right angle. And this is how infinitesimal calculus works (although I’m not very good at calculus at all).

      • Zaphod says:

        An interesting thing about Pi, it contains your phone number. Mine too. Everybody’s, in fact.

        It also contains next weeks winning lottery numbers, in the order that they will be drawn.

        It would be very strange indeed if it didn’t, of course, as it’s infinite and non-repeating. Every possible string of numbers is in there.
        To understand this better, it’s easier to consider the odds of a particular sequence not being in there. Start by considering a string of three. Then just keep expanding.

        Infinity is very odd. (Or maybe it’s even?)

        • Frank Davis says:

          sounds a bit like Borges’ Library of Babel.

        • Tony says:

          Sorry to be the fly in the ointment but pi does not need to contain every phone number or other sequence. Although it might do. Infinity is even stranger than you say. Let me quickly construct an infinite number like pi that is missing the number 3.

          Just start with the infinite decimal expansion of pi and then cross out all the 3s. Simples.
          Ok, strictly speaking, that does require a full proof but I hope it illustrates the principle.

          Technical note, the above works with all ‘Irrational’ numbers, whether ‘Transcendental’ (like pi) or not. The only exception would be an irrational number constructed by adding 3s randomly to the decimal expansion of a ‘Rational’ number

        • Zaphod says:

          Tony’s number (below) is infinite, but it’s not random. :-)
          We could now consider the real meaning of random?

        • Tony says:

          The real meaning of random ” – A seriously deep and difficult philosophical question.

          Given our total reliance on secure IT these days, it could even be described as the single most important question the human race faces this century. Although that might be slightly over stating the case.

          My best take on it is to simply say that true randomness has never knowingly been observed.

  2. Zaphod says:

    Allegedly, it was “experts” who once proclaimed that Pi would henceforth be exactly three.
    For the public good.
    They may not have been correct, but they had the power to enforce that belief.

    If this story isn’t true, then it should be.

    • Frank Davis says:

      The ancient Egyptians said that pi was equal to 3. But since I was able to find a more accurate value than this in 10 minutes, using just a piece of string and a ruler and a circular pot (all things they had too), I’m sure they would have been able to do exactly the same as I did.

      But I don’t think they used decimal numbers like I was. Maybe they didn’t even have fractions. 3 and 1/6th would have been more accurate than 3.

  3. woodsy42 says:

    In normal daily life absolute precision rarely matters. What matters is a clear approximation sufficiently accurate to make things work or to compare different outcomes. The figures being input to the question are often already only approximations or assumptions.

    • Ripper says:

      There is no such thing as absolute precision, it goes on and on, like pi. Way back in the 70s I operated a machine known as an Optical Profile Grinder. This used the shadow of a magnified drawing to accurately grind the profile of cutting tools. It was accurate to +/- 0.0001″ and I remember thinking to myself, what is the point of having such accuracy when ambient temperature would expand or contract the tool one hundred times that much. All we can ever do is make things as precise as we are able to, and that is the way it will always be. The molecule is the limiting factor.

      Take a simple thing such as a razor blade, for example. The edge needs to be ground very accurately and it takes precision grinders to do that, with vernier adjustments. Yet items from the stone age such as flint axes and arrow heads have been found, where the edge is just one molecule thick. We go around in circles searching for perfection and it does not exist.

    • beobrigitte says:

      There is no such thing as absolute precision, it goes on and on, like pi.

      True. For something to make sense you need to combine precision with accuracy.

  4. Rose says:

    That took them long enough to work out.

    American initiative to stop people smoking coincides with rise in obesity
    5th June 2016

    “In order to approximate the effect of various socio-environmental factors — things such as changes in food prices, physical demands at work, urban sprawl, racial composition, age distribution and cigarette consumption — Baum and Chou used almost 30 years’ worth of data from the National Longitudinal Survey of Youth. The survey gathered detailed information about the social and economic backgrounds, weight, height and many other characteristics of more than 12,000 youth. The researchers controlled for several variables, including age, education, income and work experience. And what they found is that nothing seemed to have had much of an effect at all. That is, aside from the changes in cigarette consumption.”

    “The impact associated with the fall in cigarette consumption was the largest of all the factors the researchers tested — such as the rise of urbanization, the fall in national food stamp enrollment and the growth in the number of restaurants — and that held true in all three of the models they built to compare the various factors.”

    • Joe L. says:

      Long enough indeed. However, don’t think for one second that Big Pharma hadn’t researched this thoroughly years ago; this was their wet dream. Denormalize both smoking and obesity, then scare people into buying NRT and antidepressants to quit smoking, then once they become obese, scare them into adding on another cocktail of drugs to help them acheive ‘normal’ weight. I only wonder what their next goal in this pyramid scheme of ‘health’ is.

    • waltc says:

      They’ve actually known this for a long time, at least in the US. From at least ten yrs ago I have a chart showing the inverse percent of smokers to fatties running from the 1980s to some point in the 2000s. Then too a CDC chart inadvertently told the same story. It’s been the elephant in the room for a long time

      • Some French bloke says:

        They’ve actually known this for a long time

        Absolutely, to say nothing about the impact of decreasing smoking rates on the soaring rates of degenerative conditions such as Parkinson’s and Alzheimer’s diseases. Cassius Clay’s rather early demise (“I don’t smoke but I keep a match box in my pocket. When my heart slips towards sin, I burn a match stick and heat my palm with it.”) is a case in point.

  5. junican says:

    There is no doubt that the circumference of a circle is directly related to its radius. And yet we are unable to find a mathematical equation which which give an exact quantity as its result. Weird or what? I sometimes wonder it the reason has something to do with the curvature of the universe. For example, could you curve a circle in such a way that the radius is a tiny bit longer, so as too have a precise radius for a given circumference. I am thinking of something like a upturned saucer. I must admit that I have no idea of what the maths of such a figure would look like.

    • woodsy42 says:

      So do you mean that pi really is a simple exact number (like 3) but the curvature of space screws up our measurement?
      If so I rather like the idea.

      • junican says:

        Well, yes, exactly. The results of trying to measure pi on a totally flat surface result in nonsensical infinities. Does that not suggest that flat surfaces do not exist in nature? But there is a problem, which is the nature of spheres. A perfect sphere also has an infinitely variable radius as compared with the surface area of the sphere – or vice-versa. A perfect sphere MUST have a perfect relationship between the radius and the surface. So why is pi indeterminable precisely? It can only be that our maths do not accord with reality.

        • west2 says:

          Interesting thoughts and ideas that mathematicians have grappled with for ages.

          Once you go from 2d space into 3d space you move from Euclidean Geometry to Non-Euclidean geometry, You then need the help of Riemann, Gauss and others. The maths becomes very complex.

          As you suggest there maybe spaces where the ratio is different. That means for some spaces the ratio is no longer Pi. With different spaces Pi doesn’t change as it is a mathematical constant.

        • Frank Davis says:

          Once you go from 2d space into 3d space you move from Euclidean Geometry to Non-Euclidean geometry,

          Really? My orbital simulation model uses Euclidean Geometry (it’s the only kind I understand), and works pretty well. And so far no astronomer has suggested that its inaccuracies are due to my use of this geometry.

          But then, my model doesn’t include any relativistic effects: gravitational forces act instantaneously between sun and planets.

    • Frank Davis says:

      Isn’t there a mathematics of curved space? Riemann?

      • west2 says:

        Yes, Riemann as I suggested in my reply to Junican.

        You could use one of a variety of geometries. Using Euclidean geometry helps simplify the problem though may have trade offs and limitations, as you suggest. In many uses the trade offs are worth it.

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