For most of us, it is intuitively clear what we mean by measuring a length or a distance – but what, really, IS an angle, and how should we measure it? Many are flustered, if put on the spot.

We know from experience that the number we come up with for the measurement of length depends on the ‘units’ we use to measure – a table top may be 140 centimetres, 1.4 metres, or 1400mm long, and depending on the industry we are working in, there may be standard expectations. Even though this is a well known and much repeated point in physics education, the team looking after NASA’s Mars Climate Orbiter miscommunicated about units and the craft burned up in the Mars atmosphere.

Most of us think of angles as being measured in units called ‘degrees’ – 360 of which make up a full revolution. In fact, physicists don’t like to think of angles as having real ‘units’ in the usual sense. For measuring length, you really have no choice but to choose a specific set of units, and there really is no sense in which there are correct or incorrect choices – it is just a matter of preference. The same applies to mass, and to time. The combination of metres for length, seconds for time, and kilograms for mass is a common choice known as the ‘mks system’ of units. And while there are standard add-ons to cater, among other things, for electric charge (or, equivalently, electric current – which incidentally also includes everything we could want to say about magnets) and temperature (please not Fahrenheit) – there is no add-on to MKS for the purpose of discussing units of angles.

The failure to make a special fuss about angles is not due to angles being unimportant. Angles matter a great deal in how we understand spatial arrangements and interactions of objects. Note that they matter in understanding ‘spatial’ matters. This is a separate domain from such things as mass and electric charge, which are fundamentally different kinds of things, which don’t have anything to do with spatial arrangement as such.

One of the most important and recurring themes in all experimental / observational science – perhaps the most important one – is the notion of scale. Is something big or small? Do we have a little bit of something, or a lot of something? Is one thing bigger than another? And a crucial thing is that whether something is ‘big’ or ‘small’ has no intrinsic meaning – it is only big or small when compared to something else that is meaningful in some context. A molecule is small compared to a cell, and a cell is small compared to a person, but a molecule is huge compared to a proton. When we ‘zoom’ in or out with our microscope or telescope, or our timeline, things look different in a well defined way that we understand. Scale is not mysterious.

There is a wonderful puzzle about scale, which I heard on one of the great sources of puzzles – the (erstwhile) long running show on US National Public Radio called ‘Car Talk’ – well known to be about much more than just cars. In effect, the question was: what can you draw on a piece of paper that will look the same when viewed just so by eye, as when viewed through a magnifying glass?

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There are a number of answers, including colouring in the whole page, or drawing a straight line across the page and colouring in one side of the line. Of course we shouldn’t draw a line which makes a small triangle across one corner of the page, because that small triangle would look different under a magnifying glass.

The cute general answer is something like this – we can colour in any shape made up of straight edges, as long as the distance between corners is large enough that we can’t see more than two adjacent edges at a time through the magnifying class. The point is that angles don’t look different under a magnifying glass – angles don’t scale.

One second is still one second, and one kilogram is still one kilogram, whether you are using a magnifying glass or not – but this seems rather trivial because we know that time and mass are fundamentally different from length. Angles, on the other hand, are a geometrical idea, but angles don’t rescale either. A right angle is a right angle no matter how much we zoom in or out, and a 30 degree angle remains a 30 degree angle.

So, how DO we define a measurement of angle? There are different ways of approaching it. One could think of constructing (equal) wedges, like slices of a circular cake, to define fractions of a circle, but this gets cumbersome and still hides concepts that it would be useful to spell out. The slice idea, however, does give the useful intuition that an angle can be thought of as a fraction of a circle – and that is an important point, into which we can dive more deeply another time – perhaps on the 28^th of June, …

It’s a recurring theme that things which don’t need explicit units to be measured are most often a ratio of one number and another, where both have the same units. For example, the ‘aspect ratio’ of a rectangle is the ratio of its longer length to its shorter length. If you make a rectangle out of two squares, the aspect ratio is 2. The ‘golden’ ratio is about 1.618 (it’s cute – you can read further in many places). Whether you measure the rectangle’s sides in microns or lightyears, the aspect ratio is the same.

This idea, that ratios make good measurements of things that in some sense should not depend on our underlying units, this suggests a nice definition of an angle. Think of a slice of a circle defined by two radii (plural of radius) of a circle. There are two lengths we can see here – 1) the length of the radii of the circle (which are of course the same as each other given what a circle fundamentally IS; and 2) the length of the curved piece of the slice – which is some fraction of the circumference of the circle, just as the angle is this very fraction of a full rotation. We have a clear choice: 1) define angle as the fraction of a full rotation; or 2) define the angle as the ratio of arc-length S to radius r.

There is a longer story which indicates we can almost have it both ways – but the shorter story is that is turns out to be most convenient to define the measurement of an angle as the ratio of arc-length to radius – which is incidentally the same, for a particular ‘angle’, no matter whether we choose to work with a circle of small radius to large radius. One interesting way to understand how things like this fail to depend on the overall size of the picture (or other physical representation) which we are using is to notice that the numbers change the same way whether we: 1) double the size of the picture or other model of our scenario; or 2) use a new unit of length which is half as big as the one used previously.

Another way of saying this, then, is to use a reference circle whose radius is 1 unit of whatever we are using for measuring length. Then the magnitude of an angle which defines a slice is given by the arclength of the curved piece of the slice. This ratio: the number you get when you divide the arc length by the radius, is formally ‘unit-less’’ (often called ‘dimensionless’, just to confuse people) but sometimes it is spoken about as if it had units – which are called ‘radians’.

So how many radians make a full rotation? You know doubt know that the famous constant known by the name Pi, pronounced like pie, and usually represented by the Greek letter π, is the ratio of the circumference of a circle to the diameter of a circle. It’s somewhat of a pity that this is the definition, because the diameter is twice the radius, and this means that a full rotation is twice π radians. So alas, the angle of a full rotation, or a whole pie, measured in radians, is not π, but 2 times π – talk about a missed opportunity!

This factor of two has caused endless algebraic mistakes, general irritation, and heated debate about why we did not make a celebrity out of the number whose value is twice the value of π, but the details of that debate are matter for the 28^th of June.