$\text{Angle}=\dfrac{\text{Arc length}}{\text{Radius}}$, a result or a definition?

Question

There is a commonly used formula to find an angle $$ \text{Angle}=\dfrac{\text{Arc length}}{\text{Radius}}. $$

My question is whether this is a deduced formula or it is the very definition of an angle? If this is the definition, why does this make sense? If this is formula, what is the proof?

Thanks!

Answer 1

It depends. You can define a radian as the angle subtended by om the centre of a circle which intercepts an arc equal in length to the radius of the circle. Or Equivalently you can say that total angle of one complete revolution is $ 2 \pi$.

In second case you can derive your formula from the definition, while in first it is a part of definition.

Answer 2

When defining how to measure something (for example mass), one has to define a unit (obsolete example “a kilogram is the mass of this metal cylinder”) and what it means to be a multiple (e.g., of limited use: “the weight of an x-fold mass deforms a spring by x-fold amount”).

With (the measure of) angles, we want congruent angles to have same measure and additive Ty for juxtaposed angles. This alone won’t take us far: We might deduce that a right angle has $\frac14$ of the measure of the full angle, and that the interior angle of an equilateral triangle is $\frac23$ of a right angle. Irrational angles would however be still allowed to have arbitrary measures until we additionally impose an order relation between angles in the obvious way - but is it really obvious?

We can reduce angle measurements to area measurements by fixing a circle and declaring the measure of an angle as (a fixed multiple of) the area of the “pie piece” cut out by the angle when it’s apex is in the circle centre. By rotation invariance of the circle, congruent angles have same measure, as desired. We have additivity as desired. So this would give us a nice definition of angle measurement, though a strict definition of area of a pie piece may require us to know about integration. Then again, a rigorous definition of arc length is also not that trivial.

At any rate, the SI unit radian is defined exactly in that way via arc length over radius.