Is angle an instance of something more abstract than angle? Is an angle (as generally understood) best described as a relation or a quantity?
 A: If you think of a quantity as something like a distance between two points, there are some similarities with angles, for they also have a numerical value, but there are also some differences.
First of all, an angle is usually defined as the length of a fraction of a circumference with the angle's vertex in it's center, but you can't take measurements of the arc of circumference without setting the value of the radius. One approach to this problem is to give the radius a standard value of 1, but this is another way of saying that the arc must be divided by the radius (as they are proportional).
As the value of an angle is the result of a division of lengths, it is rather pointless to give it units of measurement, because those units would be something like meters/meter or inches/inch (just like the scale of maps doesn't have units). The radians and the degrees are what you may call "adimensional units".
Therefore, angles are quantities to the extent that they have a numerical value, but you may consider them relations because they are ratios.
