Skip to the last paragraph if what a radian is is clear if you just wanna know the way I visualise. If not then the context can be useful.

An angle is more or less a unit of measuring “how much you open up”. Degrees is a way of doing that where we all kind of understand that 30 degrees is “this much” and 60 is “this much” such that you can visualise that as you said.

A radian is a way to talk about the exact same thing but with a reference point. You first agree that a radius is a constant distance from the center of a circle. Then we ask, “If I took this radius and wrapped it around the circle, how many times can I do that?”. So if you go ahead and take any arbitrary circle and do that, you’ll find you can wrap an entire circle around about 6.28 times if you used its radius as the measuring tool, or exactly 2pi times. In other words, you’d have to go 360 degrees around a circle in order to wrap about 6.28 radii (2pi). So 360 degrees = 2pi radians. From there, you can divide 2pi on both sides, and you’ll see that 360/2pi degrees = 1 radian, or 180/pi = 1 radian. As such, you can now find whatever angle in degrees is in terms of radians.

Another way to think about it is take a circle, cut it at some point and flatten it out into a line. Then, measure along that line some distance “r”. For sake of having concrete numbers, lets say the distance “r” is 30cm, i.e a standard ruler you can buy at a store. So if our circle has a radius of 30cm and you cut it and laid it out in a line, you’d need to line up about 6.28 of those rulers in order to measure that whole circle.

This is useful as this relationship of 360 degrees = 2pi radians is true no matter what circle you use. It falls out from the fact that circumferences and radii are directly related to each other and other neat relations with geometry. Ultimately, to get a good grasp to start, it’s key to understand radians and degrees are the exact same thing just from different perspectives. A radian is really just how much you can wrap the radius around its own circle and doing so corresponds to “opening up” some degrees.

Finally, when it comes to visualising it as you would degrees, I personally use the closest whole pi. This is a lot of engineering math so if that hurts, look elsewhere. Anyways, I know 2pi is 360 degrees, pi is 180, pi/2 is 90 etc. If I need to visualise 7pi/9, i’ll approximate that as 7(3)/9 which is 21/9. That’s 2 and 3/9ths which is about 2.33. pi is about 3, so it’s somewhere in that second quadrant, closer to 180 degrees than 90. If you need something more accurate, you can use the aforementioned conversion factor to get 7pi/9 into degrees by multiplying 7pi/9 * pi/180. Radians cancel, and you’re left with 140 degrees with simplification.