Measuring the (Seemingly) Impossible

(This talk was given as part of the Adventures in the Classroom lecture series at the University of Toronto on August 7, 2014. It aims to connect math educators with researchers in the academic math community. A video of this talk will be available later.)

We will introduce two fun projects involving Geodesy, which is devoted to measuring things to do with the Earth.

From Harold Llyod’s “Safety Last!”

“Without doing anything dangerous, measure the height of your school”

I love this question. It is a bit affronting (how could I possibly do that?!), but there’s something magical about it that draws you in and gets you using your imagination. If you have never thought about this question before, please take the time now. (Seriously, go for a walk and think about it.)

I have posed this question to many people and have received many different solutions. When I talk to people about this problem I am struck by what solution they think is the most “obvious” or “natural” solution. Think about that as we examine this problem.

Let’s get straight into some solutions that I’ve heard.

The “Math” solution

The idea here is to go out on a sunny day and measure the length of the shadow that an upright metre stick casts on the ground. That tells you two sides of a right triangle (the one formed by the metre stick and its shadow). Now you can measure the length of the shadow of your school. The triangle formed by the school and its shadow is in the same ratio as the triangle formed by the metre stick and its shadow (they are similar triangles). Using the fact that their ratios are the same you can solve for the height of the school.

This is essentially illustrated here.

Credit to Cathleen Sanders for the useful picture. Visit her site for more information on this project.

This seems like a fairly nice solution. It is easy to execute, scales nicely to larger structures and the underlying math is fairly elementary. However, I don’t think this is the first solution we should be teaching students. I think the solution is nice and beautiful and satisfying, but it is too alien for a first solution. Let me explain.

This solution introduces many surprising extra elements

Why are we using shadows? Why are we using similar triangles? What part of the question suggests that this is one of the ways to approach it?

Students should not have too much trouble understanding this solution, but remembering this solution is a whole different story (“It was something about shadows I think…”). This solution is clever, but seems to just appear out of nowhere. Part of the beauty of this question is there are dozens of possible solutions, and I feel like this one stifles problem solving because students begin to focus on geometry, rather than the task at hand.

That being said, my wife argued (fairly persuasively) that this is a  natural solution for anyone who has seen geometry recently. So perhaps my complaint is less valid.

A method for getting new solutions

I could, if I wanted, just list out possible solutions to this. Perhaps I could dazzle you with the variety and creativity of my solutions. That isn’t the goal though. I’m going to show you a handy method for problem solving that naturally leads to many possible solutions.

Identify the key difficulties

Try to identify what part of the problem is giving you trouble. You can state it precisely, or abstract a little bit. Here are some possible difficulties you might encounter in this problem:

  1. “The wall is too big.”
  2. “I can’t get to the top of the wall.”
  3. “Large vertical distances are hard to measure.”

Ask related questions

Let’s try to break down these difficulties with related questions:

  1. Is the tallness of the school really the problem? Are there other big things that you know how to measure? For example the school yard is much bigger than your school and you can probably measure the yard. What is the real difficulty?
  2. Do you need to get to the top of the wall, or do you just need something to get to the top of the wall? Can you send something else in your place?
  3. What sort of distances are easier to measure?

Answer these related questions

For questions 1 and 3 we can see that large horizontal distances are fairly easy to measure.

The “spinny-clicky thing” also known as a Trundle Wheel


So our problem isn’t measuring large distances, it is measuring large vertical distances.

So… how can we get around this? Well let’s try the most naive thing we can “turn vertical distances into horizontal distances.” This sounds tricky, but let’s see what sort of things we can come up with. By the way, turning something hard into something easy is one of the key problem solving techniques. Frame what you don’t know in terms of things you do know.

Some (possibly dangerous) ways to turn vertical into horizontal

  • Knock down the wall. (Well, this could work, but there are all sorts of logistical problems with it. What would happen if we actually did it though? Really we would be doing the next thing…)
  • Count the number of bricks in the wall, then lay a couple bricks on the ground and measure them.
  • (Can you think of any others?)

That’s neat! We realized that a dangerous solution can actually lead to a simple, non-dangerous solution!

Back to question 2 – Do you need to get to the top of the wall?

What are some ways that we know to get things on to the top of the school? Or, more generally, how can we overcome large vertical distances?

Here are a couple of the ways that I’ve heard:

  • Rockets or Catapults. (Not so crazy of an idea!)
  • Fire Trucks.
  • Stairs.
  • Strong Muscles, or throwing something.
  • Slingshots.
  • Tall animals (Giraffes?)
  • Birds.
  • Animals that climb walls. (Ants, squirrels, …)
  • Trees.
  • Smoke or Hot Air.
  • Helium Balloons.
  • Laser Pointers. (In general: light.)
  • Sound. (Echos)
  • Rain or Snow. (These overcome vertical distances in the other direction, from the sky to the ground.)

The neat thing is that most of these tools will give solutions to our problem.

Use these tools to solve the problem

For example, let’s take helium balloons. It seems fairly natural that we could tie a long string to a helium balloon, let the balloon float to the top of the school, then measure the length of string needed.

Or let’s take strong muscles. We can throw a ball attached to some string on to the school, mark off the string used, then pull down the ball and measure the string used.

These solutions seem genius and creative, but really they are logical and come about naturally from examining the problem. This is useful when (like most mathematicians) you aren’t particularly clever, but are good at asking questions.

Inside the Box thinking and Outside the Box thinking

Let’s take this opportunity to talk about “inside the box thinking”. It often gets a bad rap and is used to indicate complacency, submission and boringness. Outside the Box thinkers are usually those among us who are thought to be exciting, original, creative and getting promotions. I want to dispel this a little bit.

An inside the box thinker will ask the following questions when faced with a problem:

  • What is the problem?
  • What is the key difficulty?
  • Do I know ways to solve related problems?
  • What are the known tools for overcoming the key difficulties?
  • What are the rules? How do the rules interact with each other? How do I exploit them? (These are particularly true of such a thinker playing board games.)

What does an outside the box thinker try to do to solve a problem?

  • Be creative.
  • Be clever.
  • Be original.

Those things are well and good, but I don’t really know how to teach these things in a concrete way. I would also wager that most people (like me) aren’t particularly clever or creative or original, but that’s okay! I think that inside the box thinking will get you pretty far!

Example: Poker

In general mathematicians can make very good poker players. They have a good sense of probability, expected value, what hands are good, how often to keep hands, etc.. They have a strong understanding of the rules of the game and how probability interacts with them. As a result, among amateurs, mathematicians can become near the best players. You will do very well at your local poker night if you just stick to “playing the odds”. This is the inside the box approach to poker.

Will you become one of the top pros doing just this? No, probably not. To do this you need to be thinking more like Tom McEvoy’s famous quote: “Poker is not a card game with people, it is a people game with cards.” This is more like outside the box thinking. You need to see the game not merely as a card game, but a people game.

You can examine this in more detail with this article “Why Intelligent People Sometimes Struggle with Poker”.

Summary of the problem

Before we move on, let’s summarize our work on this problem:

  • We identified the key difficulties. (Overcoming vertical distances.)
  • We asked related questions. (What things can overcome these?)
  • We answered these questions. (Light, Fire Trucks, Trees, Helium Balloons)
  • We adapt (some of) these tools to solve our original question. (Let the Helium Balloon float to the top of the school, measure the string used.)

Teaching this project

When teaching this project you are likely to find students who get overwhelmed by the question. The massiveness of the problem is part of its appeal, but some for some students this is very off-putting.

To try to counteract this help them break up the problem into more manageable bites. Use the approach we’ve described to allow the student to engage in questions on a smaller scale. Encourage the student to be analytic, rather than creative if they find themselves being overwhelmed; being clever is hard, identifying what is giving you difficulty is liberating.

Some students will also latch on to different aspects of the problem. Some will like the brainstorming (the strategists), some will like the building/execution (the engineers), some will like the teamwork.

Going Further

What to do with students who need a bigger challenge? One of the most straightforward things is to ask for measurements of bigger things! In Toronto, the natural thing is the CN Tower. How’s that for cool? What student will forget about trigonometry once they’ve used it to measure the height of the CN Tower?

A word of warning though, you need to exercise judgement when giving this problem. For a student who barely struggled through the school measuring problem it can be quite disparaging to get this problem and find that most of your old tools don’t work. This problem is less forgiving than the previous one.

That being said, all of a sudden things that were easy become hard. For example, is it easy to find the end of the CN Tower’s shadow? This on its own could be tricky!

Other problems in Geodesy

Let’s mention some other problems that go further than this, but are still within the grasp of (motivated) students.

  • What is the circumference of the Earth?
  • How far away is the moon/sun?
  • How big are they?
  • How far away is your favourite star?

These problems should all be solvable in theory with the ideas available to students. In some cases the limiting factor is the accuracy of the physical tools available.

Let’s focus on one of these problems…

What is the circumference of the Earth?

Let’s focus on three nice historical solutions to this problem (although there are other solutions I’ve omitted). Most of this was gleaned from the Wikipedia article “History of Geodesy”, which is a fascinating read!

~250 BCE – Eratosthenes’ Shadows

This is probably the most famous method. It basically amounts to measuring the shadow of a metre stick in two places at the same time, when you know that one of sticks has no shadow at that particular time. In that case, knowing the angle created by the shadow in the place where there was a shadow (in Eratosthenes’ case it was one fiftieth of a circle) and knowing the distance between the two measurement sites (in this case 5000 stadia), is enough to tell you the circumference of the earth (in this case 50 times 5000 stadia).

Here’s the picture:

Taken from Wikimedia Commons.

~830 CE – Travelling to Mecca

This next one is really neat because it is so culturally motivated. Suppose for the moment that you are a Muslim in the 9th century. It is important to you that you know which direction Mecca is in (so you know in which direction to pray) and it is important to you that you know how far away Mecca is from your home (so that when you make your pilgrimage you know how many supplies to bring, how long it will take, etc.). It turns out that these measurements, together with the development of spherical geometry, led to an estimate for the circumference of the Earth.

From Wikipedia:

Around AD 830 Caliph al-Ma’mun commissioned a group of astronomers to measure the distance from Tadmur (Palmyra) to al-Raqqah, in modern Syria. They found the cities to be separated by one degree of latitude and the distance between them to be 6623 miles and thus calculated the Earth’s circumference to be 24,000 miles. Another estimate given was 5623 Arabic miles per degree, which corresponds to 111.8 km per degree and a circumference of 40,248 km, very close to the currently modern values of 111.3 km per degree and 40,068 km circumference, respectively.

990 CE – The 17 year old

You should read about the exploits of Abu Rayhan Biruni here. At 17 years old he came up with a new method for estimating the circumference of the Earth that avoided “walking across hot, dusty deserts”. The method is nice and is within the grasp of a high school student.


I hope you enjoyed this foray into Geodesy. I certainly enjoyed learning the interesting history of Geodesy (like the Principal Triangulation of Britain and the Retriangulation of Great Britain.)

Please let me know if you try any of these experiments on your own; I would love to hear about them or see pictures! You can get in touch with me here.