Sunset along the Milwaukee River

Degrees Versus Radians. What's the Difference? Why Learn Both? What About Dungs Dungs?

Date: 7/15/2026

Level of difficulty: High School/College

First question, what is a degree? We know a degree is a unit of measurement, but what exactly does a degree measure? Just now to help me try and make sense of it, I held up my pen parallel with the floor, kept the left end centered (it did not move), and started rotating the right end up and around. Each time I stop somewhere in this process, the amount of rotation is different. For example, when I am simply holding the pen in my hand parallel to the floor, there is zero rotation, but the moment the right end of the pen starts to travel up and around, there is clearly an amount of rotation occurring, and that is precisely why we invented this unit of degrees to try and capture that amount.

For units of measurement, such as degrees, to make any practical sense to us, they need useful physical reference points which we can then assign numbers to (numbers of our own choosing by the way). To that end, let us consider two amounts of rotation. One good place to start is an object that does not rotate (in our example, this would be the amount where the right end of the pen does not move at all, it simply stays parallel with the ground). We need to assign a number to capture this amount of rotation. Technically, there is not a right or wrong answer, we would just need to remember that whatever number we assign refers to the physical reality of zero rotation. However, practically speaking, most humans would agree and settle on the number zero to describe an object which exhibits no rotation (in our case, the pen did not rotate so we can call that amount “0 degrees”). But now we need a second physical reference point that captures another amount of rotation. One natural amount is the amount where an object rotates all the way around completing one full rotation (in our case, that would correspond to rotating the right end of the pen all the way around until it is back where it started). Now we need another number to capture this specific amount of rotation. Unlike the first amount that had zero rotation and more or less is universally agreed upon to be called “0 degrees”, the number we assign to one full rotation is not as clear. If we surveyed thousands of kids who know nothing about trigonometry what number to assign to one full rotation, I wonder how varied, understandably, the responses would be. I suspect we may see answers like 1 or 100, and of course random numbers like 423342. And importantly, none of their answers would technically be wrong, because it is we humans who invent units of measurement to make sense of the world. They are not things set in stone. The ones we see today are the ones we just agreed to, but maybe on some other planet, the alien adults agreed on other definitions of units to make sense of measurable things in their world (hotness/coldness, heaviness/lightness, tall/short, etc.).

Before assigning a number to this amount of rotation, notice how these two reference points (zero rotation and a complete rotation), and everything in between them, captures the entire spectrum of how much something can rotate. We could think in these terms,

1 full rotation = some number x

½ of a rotation = ½ of that number x

¼ of a rotation = ¼ of that number x

1/100 of a rotation = 1/100 of that number x

34/98 of a rotation = 34/98 of that number x

and so on…

Now most of us, without probably ever questioning why, just automatically assume 1 full rotation = 360 degrees. It’s not right or wrong, it’s just the number humans agreed upon for this made up unit “degree” to capture 1 full rotation. Maybe at the dinner discussion years ago while debating what number to assign for this unit of degrees, instead of someone shouting 360, someone could have shouted 500 and the math would be equally as sound. For example, “250 degrees” would correspond to half of a rotation (instead of 180 degrees), “125 degrees” would correspond to a quarter of a rotation (instead of 90 degrees), and so on. The main thing to appreciate is that the number we traditionally use, 360, to designate a full rotation is completely arbitrary. It comes from ancient Babylonian mathematics and astronomy where they approximated 1 year to take 360 days (ie 360 days for the earth to complete 1 rotation around the sun).

So we know 0 degrees corresponds to no rotation, and 360 degrees corresponds to 1 full rotation. Now a good question is, what does “1 degree” actually mean in terms of rotation? To make it concrete, what would “1 degree” of rotation mean in terms of our pen rotating, what would that actually look like? One way to think about it is that it’s the amount such that if rotated that same amount 359 more times, the pen would complete 1 full rotation. Take a moment to really think about this until you’re convinced of its truth. Admittedly, 1 degree is a bit hard to mimic with extreme precision on a pen, but what about trying to to make sense of “90 degrees”, what would the pen’s rotation look like then? Well, one way to think about “90 degrees” is that you have ninety 1 degrees, in other words, we repeat the amount of rotation for 1 degree ninety times (still a bit hard to be precise). Another interpretation of what “90 degrees” looks like, is to consider what amount 90 is of 360 (ie 90/360). The reason behind asking this is that I know 360 degrees = 1 full rotation, now if I can make the left hand-side of that equation equal 90 degrees somehow, I will simultaneously know the amount of rotation that number corresponds to. Using basic algebra, 360*x = 90 => x = ¼; to maintain the balance of the equation we have: 360*¼ degrees = ¼ of 1 full rotation (ie 90 degrees = ¼ of 1 full rotation). As one last example, let’s try to make sense of what “120 degrees” looks like in terms of the amount my pen should rotate. Using the same strategy, I know 360 degrees = 1 full rotation, but I am investigating 120 degrees and need to know what amount that is of 360, therefore we have 360*x = 120 => x = 1/3; maintaining the balance of that core relationship we have between degrees and rotation we have: 360*1/3 = 1/3 of 1 full rotation (ie 120 degrees = 1/3 of 1 full rotation). We can apply this understanding to make sense of any amount of degrees and the corresponding amount of rotation it represents.

Now let’s move onto the dinner table next door where instead of someone shouting “360 degrees should represent 1 full rotation”, at this particular gathering someone shouted “2π radians should represent 1 full rotation”. What on earth is a radian and why 2π? Hopefully by now you know the term “radian” is completely made up and the choice of the number 2π is completely arbitrary (it is related to the fact that the circumference of a unit circle is 2π units). What we need to keep coming back to is the physical interpretation of what 1 full rotation looks like regardless of where you are in the universe, the number and made-up word we assign to that universal physical reality are arbitrary afterthoughts. In fact, our reasoning above about “degrees” will be the exact same when we reason about “radians”. Instead of reasoning with the relationship 360 degrees = 1 full rotation, we’ll instead use this newly defined relationship 2π radians = 1 full rotation. So what does it look like if my pen rotates “π radians” for example? Using the same reasoning we employed with degrees, we need to determine what amount π  radians is of 2π radians. Using basic algebra, 2π* x = π => x = ½; maintaining the balance of that core relationship we have between radians and rotation we have: 2π * ½ = ½ of 1 full rotation (ie π radians = ½ of 1 full rotation).

What about “1 radian”, what does that look like in terms of rotation? Well, we need to determine the amount of 1 radians in 2π radians => 2π * x = 1 => x = 1/2π. Maintaining the balance of that core relationship we have between radians and rotation we have: 2π * 1/2π = 1/2π of 1 full rotation (ie 1 radian = 1/2π of 1 full rotation). This may seem a touch complicated, but if we consider that the number 1/2π, is in fact just a number that roughly equals .16, this .16 can simply be interpreted as a 16% rotation (which most of us can have a rough estimate of what that would actually look like).

So is there anything special about “degrees” or “radians”? Short answer is no. At a third dinner table, someone could’ve shouted “2880 dungs dungs should represent 1 full rotation”, and generations to follow would unquestionably answer questions about dungs dungs (that logically are equivalent to degrees and radians). What personally helps me make sense of these units is to continually remind myself that these made up terms and arbitrarily assigned numbers refer to some concrete physical reality. With all that said, hopefully you are thoroughly convinced that sin(240 dungs dungs) = ½…

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