# Derivation of the length of an arc formula

My textbook says that the radian measure of an angle is the ratio: $\theta = \frac{s}{r}$ Where s is a portion of the entire circumference, and r is the radius. So essentially the arc length is thus: $s = r\theta$

However instead of showing the length of an arc formula this way it says:

$\frac{{{\text{Length of arc}}}}{{{\text{Circumference}}}} = \frac{{{\text{Angle AOB}}}}{{{\text{Total angle around O}}}}$

(where arc AB has length s)

so that gives:

$$\frac{s}{{2\pi r}} = \frac{\theta }{{2\pi }}$$

which cancels to give:

$$s = r\theta$$

Okay i understand that, but my question is (probably a dumb one) what is the reasoning behind the equivalence relationship: $$\frac{s}{{2\pi r}} = \frac{\theta }{{2\pi }}$$

How is it that we can equal the two together when one side is concerned with lengths (length of arc/length of circumference) and the other is concerned with angles (angle aob/total angle around O).

If someone could give me an intuitive answer i'd love that, thank you.

• Do you know the circumference of the circle and the total angle around $O$? If you do, then you get that expression. Finally, simplify it to obtain the length of an arc equation. What this proof shows is that the larger the angle, the larger the length of the arc. Thus, we can use the ratio of a certain angle and total angle and the ratio of length of arc and circumference. Therefore, we have $s = r\theta$ – NasuSama Jan 5 '14 at 22:36

You are right. This looks like the argument is backwards. Your interpretation is correct and from it you derive the one full turn is $2\pi$ radians.