Radians to degrees made simple

I was thinking about radians today and  how they are annoying and unintuitive to deal with. After thinking about it for a while I realized that when you look at radians in a certain way, there's an emerging pattern which makes radians intuitive, fun, and easy.

You know how we're taught about radians that:

2π radians = 360°
π radians = 180°

And so on. And you can do your degree conversions like so:

 π radians
-----------  = 1°
    180

And it's all right and well, but... radians can sometimes be kind of hard to keep in your head when you do more complex conversions.

And if you're like me, it's easy to lose track of what you're doing and what the numbers actually represent. You end up going back and forth and becoming confused.

Don't worry. There's an easier way to visualize radian conversions.

Take a look at this:

2 / 1 * π radians = 360°
2 / 2 * π radians = 180°
2 / 3 * π radians = 120°
2 / 4 * π radians = 90°
2 / 5 * π radians = 72°
...

When we always divide two by parts-of-a-circle we get a really nice uniform and intuitive progression of numbers.

2 / 1 is of course equal to 2, so that would be the 2π radians, giving us 360°.

2/2 is of course equal to 1, ie. 180°.

The only reason I even put them there is to make the whole thing uniform and easy to visualize.

Notice how the divisor starts from 1 and goes to 2, 3, 4, 5 ... and so on. You can think of the divisor as "parts of the circle". Two times π is a full circle, 360°. When your dividend is always two, like in the above example, the divisor then always directly represent into how many parts the circle has been divided into.

1 = full circle, 360°.
2 = divided into two parts, 180°.
3 = divided into three parts, 120°.
4 = divided into four parts, 90°.
...

And so on. So, how many radians are in one degree? Easy. There are 360 degrees in a full rotation, so we just drop in 360:

2 / 360 * π radians = 1°

Why does it work you ask?

Since there are two π radians in a full circle, dividing two by how many degrees we want we get a multiplier for the conversion and everything stays nice and uniform.

In my opinion this approach makes it much easier to visualize radians to degrees conversions than using "π radians =  180°" as the starting point.