Connection between angle and arc length

Question

I read somewhere that the angle corresponds arc length in the unit circle. In other word if we have a circle with center placed on the origin and have radius equal to $1$ then length of the circular arc from point $(\cos t, \sin t)$ to point$(1,0)$ is correspond to angle. It doesn't make sense to me.

For example It is obvious to me if we represent an angle with degree, for instance $60^\circ$ the angle correspond to ratio of it to $360^\circ$ or it is $\frac 16$ of the whole circle. I know it is not precise statement but at least It make sense to me. But in the case above, I can't make a connection between arc length of the unit circle and angle, because on the one hand we have the length but on the other hand we have angle.

Answer 1

You are correct. To make the connection, we have to make the radius of the circle constant. Then, angle is defined as $\frac{\text{arc length}}{\text{radius}}$. Interestingly, both have dimensions of length, making the measure of angle unitless.

Answer 2

Note that arcs of equal length subtends equal angle at the center of a circle. Thus a function, $f:[0,2\pi]\to[0,2\pi]$, mapping a real number $\ell$ to the angle subtended at the center of the unit circle by an arc of length $\ell$, is well defined.

Can you show that this function satisifies $f(c\ell)=cf(\ell)$ for any $c\in\Bbb{R},\ell\in[0,2\pi]$ such that $c\ell\in[0,2\pi]$?

Then you know that the circumference of the circle is $2\pi$, and the considering the circle as an arc, the angle subtended by the circle at the center is $2\pi$. Hence $f(2\pi)=2\pi$. So you can deduce that $f(\theta)=\theta$ for all $\theta\in[0,2\pi]$.

Answer 3

The radius of a unit circle is $1$ m, and hence the circumference of the circle is $2\pi\times 1$m $= 2\pi$ m. Thus, for whatever angle in degrees the arc subtends to the centre, the arc length is a part of the circumference (yeah, we all know it) and using the formula for finding the angle subtended by an arc (i.e. , $\theta = \frac{l}{r}$, $l$ = arc length, $r$ = radius), you get that $\theta = l$ in radian (rather than in degrees, but the result you get is equal to the angle in degrees times $\frac{\pi}{180}$ ).

If you were not looking for an answer like this, refer @JoshuaWang's or @QED's answers.