5.1 Angles  (Page 3/29)

This brings us to our new angle measure. One radian    is the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the center of a circle by two radii. Because the total circumference equals $2\pi$ times the radius, a full circular rotation is $2\pi$ radians. So

See [link] . Note that when an angle is described without a specific unit, it refers to radian measure. For example, an angle measure of 3 indicates 3 radians. In fact, radian measure is dimensionless, since it is the quotient of a length (circumference) divided by a length (radius) and the length units cancel out.

Relating arc lengths to radius

An arc length     $s$ is the length of the curve along the arc. Just as the full circumference of a circle always has a constant ratio to the radius, the arc length produced by any given angle also has a constant relation to the radius, regardless of the length of the radius.

This ratio, called the radian measure    , is the same regardless of the radius of the circle—it depends only on the angle. This property allows us to define a measure of any angle as the ratio of the arc length $s$ to the radius $r.$ See [link] .

If $s=r,$ then $\theta =\frac{r}{r}=\text{ 1 radian}\text{.}$

To elaborate on this idea, consider two circles, one with radius 2 and the other with radius 3. Recall the circumference of a circle is $C=2\pi r,$ where $r$ is the radius. The smaller circle then has circumference $2\pi (2)=4\pi$ and the larger has circumference $2\pi (3)=6\pi .$ Now we draw a 45° angle on the two circles, as in [link] .

Notice what happens if we find the ratio of the arc length divided by the radius of the circle.

Since both ratios are $\frac{1}{4}\pi ,$ the angle measures of both circles are the same, even though the arc length and radius differ.