Back To That Envelope

“Back of envelope calculation”

I have been using estimation problems in my physics classes for more than a decade now. In fact, the first piece of writing I did for a magazine was an article in the the Institute of Physics magazine “Physics World”, was an article on this very topic. So high time to revisit it, in the light of a decade of experience.

An estimation problem is a technique to produce a rough answer to a problem, by simplifying it as far as possible, and using rough estimates for any numbers that might be needed. Physicists often call these “Fermi Problems”, after the famous scientist, who used to pose them to his graduate students. One of his famous questions was to estimate the number of left-handed piano tuners in the city of Chicago. Why are we interested in this type of problem? These days we are awash with “big data” at our fingertips, and a staggering amount of computer power to calculate an exact solution to every problem? Or are we?

These problems originate from a time when the amount of calculations was a serious limit in science. Calculations could be done by slide rules, abacus, tabulated logarithms. So to undertake a major calculation was a serious undertaking, and a quick feasibility study, an estimate, was a prudent idea, in order to see if the main calculation was worth doing. So, the thought processes needed are to simply the problem down to absolute bare essentials, and put in simple rounded numbers to get a rough answer. These are excellent analytical skills to have in many professions, not just science. Ask a painter how much paint is needed to paint your living room, or a knitter how many balls of wool are needed to complete a woolly hat. Or a chef to work out the cost of the meal, that they have put on their restaurant menu. They are all using experience and a calculation to give you their answers. They may not even be aware that they are doing a calculation, and they may not need to write anything down, but that is what they are doing.

As usual, it becomes apparent that this is not an instinctive skill, it needs to be taught and practised in order to become proficient. That is not surprising. In my introductory physics classes, I usually start off with a pirate treasure chest problem. You’ve all seen the movies — two pirates carrying a treasure chest, full of gold doubloons. Is that feasible? The answer is a resounding no. If you estimate the volume of an empty treasure chest, and the density of a metal, such as gold, a very simple calculation (seen above in the title photograph), tells you that the chest would literally weigh a ton (tons are very ambiguous, as there are several definitions, so I’m using the term for a metric ton, 1000 kilograms). Two people could not lift and carry even a modest sized chest full of gold. Arr!

I often get students to estimate volumes, because it seems, from a decade of trying, that they have difficulties with this. I often get students to estimate the volume of a bathtub in cubic metres and litres (sorry America, we use metric measurements in the rest of the world). This can prove difficult. I have seen volume estimates vary from Olympic swimming pool sized, to teaspoon sized. I even had a student tell me that they never took baths, so they couldn’t estimate the size. But with practice over several sessions, students gradually begin to get the idea and tackle some much more complicated problems, such as estimating the weight of the Eiffel Tower. In physics and engineering it is very important to get a firm grasp of quantities at an early stage. And I don;t want students to become totally dependent on calculators, computers and looking up things on the internet. These are nice to have, but not always available, and I want my students to have faith in their own abilities to function as a scientist or engineer without them.

The essential rules are:

Simplify the problem

Infer values from analogies, if the exact values are unknown

Do a simple calculation

In the case of the pirate treasure chest, assume a simple box shape. Nothing complicated. Pick an easy geometry to work with. In physics this is often a cube, rectangular box, or a sphere. There is a whole literary genre of “Assume a Spherical Cow” jokes out there. Don’t worry about the mass of the material making up the box, it will be much less than the treasure itself. Now, to calculate the mass of the gold, given the volume of the box, you need the density of gold (density is defined as mass per unit volume). You probably won’t remember the density of gold, but you might remember the density of water, which is 1 gram per cubic centimetre or 1000 kilograms per cubic metre. So just reason that a metal is a lot more dense than water, and assume a value ten times that of water. The actual value is 19.3 times as dense as water, but a factor of 1.93 is not important in this type of calculation. Now you can do a tiny bit of algebra:

density = mass / volume

so that means

mass = density x volume.

And that’s it. Not complicated. It does require some physics, particularly the definition of density, and a rough grasp of what certain values of density happen to be. Sometimes I get students to do the calculations without the aid of a calculator. The looks of shock and horror, and the audible gasps are quite entertaining, but it’s not meant to be a sadistic act of purgatory. Simply that you may have to do a calculation where there is no calculator or computer available. Suppose your smartphone (with built-in calculator) runs out of power? Do you cease to be a functioning scientist or engineer. No, you don’t. Believe that.

Now, for your homework assignment, if you are indoors, estimate the surface area of the walls in the room you are sitting in. If you are outside and reading this on a mobile device, estimate the surface area of the device. Simple. A transferable skill.