14.2. Arc Length
▪ Exploration: Try this. Locate the center of a circular object, such as a paper plate, and measure its radius. Then with a strip of paper, lay off an arc along the edge of the plate, measure it, and divide it by the radius. Next measure the central angle subtended by the arc, and convert to radians. What do you notice?
Repeat with different lengths of arc. Repeat with different sized plates. Can you make a general statement about your findings?
If we measure arc length and radius in the same units, those units will cancel when we divide one by the other. We are left with a dimensionless ratio. In your exploration, you may have seen that when dividing arc length by radius, you got a dimensionless ratio that is equal to the angle in radians. Stated as a formula,
where θ is in radians
In a circle, the central angle (in radians) is equal to the ratio of the length s of intercepted arc and the radius r of the circle.
Thus the radian is not a unit of measure like the degree or inch, although we usually carry the word "radian" or "rad" along as if it were a unit of measure.
We can use Eq. 76 to find any of the quantities θ, r, or s, Fig. 14-10, when the other two are known.
Find the angle that would intercept an arc of 27.0 ft in a circle of radius 21.0 ft.
Solution: From Eq. 76,