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I'm having a hard time understanding why

$\theta = \frac{s}{r}$

and

$s = \theta r$

where $s$ is the arc length, $r$ is the radius, and $\theta$ is the angle measure.

I understand how one is derived from the other through algebraic manipulation.

I'm trying to understand intuitively:

1) Why dividing some amount of the circle (arc length) by the length of its radius would dictate how much an angle $\theta$ "opens up." Since it's a ratio of two lengths, then why would the result be an angle measure?

2) Why multiplying the radius by an angle measure would give the arc length. I'm having a hard time grasping this one because the angle measure and the radius seem like (for lack of a better term) two different "units" to me. An angle $\theta$ is a measure of angular rotation whereas the radius is a length, so how does multiplying the two then give back another length i.e. the arc length?

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Keep in mind that, although we can assign a length to a circular arc as though it were a straight line segment, it is not a straight line segment. It bends through a certain angle with respect to the circle upon which it lies.

When we assign a measure to the arc (and thereby to the associated angle) by dividing its length by the length of the radius, we are disregarding the curvature of the arc, even though the curvature remains. The size of the ratio measures the curvature (and the angle).

It is said that the resulting unit, the radian, is a dimensionless quantity. This refers to the fact that in dividing two lengths, the unit of length "cancels out" leaving the resulting number "unit-less" or "dimensionless." The radian measure shares this aspect with other dimensionless quantities such as sine, cosine, etc.

This is not a forum on the philosophy of mathematics, but I will dare to make a philosophical point: The radian is a unit in the sense that it is a standard by which angles can be measured. It measures the angle through which a circular arc turns--the aspect of the circular arc which was disregarded when measuring it as if it were a straight line segment.

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As is well known $\pi$ is defined as the ratio between the perimeter of a circle and its diameter or 2 times the radius which is the same, so $\pi = \frac{p}{2r}$. So $p = 2\pi r$. So as $p$ is the whole circunferences then the half must be $\pi r$, $\frac{2}{3}$ of circumference must be $\frac{4}{3} \pi r$, and so on for every fraction of the perimeter. Now we defined the angle as the length divided by the radius because of this fact, because what $r$ does is just scaling the part of the circumference, but the angle remains the same.

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