College Algebra & Trigonometry, 2018a

8.1. MEASURING ANGLES 347
Measuring Angles in Radians
The other most commonly used method for measuring angles is radian measure.
Radian measure is based on the central angle of a circle. A given central angle
will trace out an arc of a particular length on the circle. The ratio of the arc length
to the radius of the circle is the angle measure in radians. The benefit of radian
measure is that it is based on a ratio of distances whereas degree measure is not.
This allows radians to be used in calculus in situations in which degree measure
would be inappropriate.
θ
r
The length of the arc intersected by the central angle is the portion of the circumference
swept out by the angle along the edge of the circle. The circumference of
θ
the circle would be 2πr, so the length of the arc would be
360
∗2πr. The ratio of
◦
θ
360
this arclength to the radius is
◦ ∗2πr
r
or
2π
360 ◦ ∗ θ
or in reduced form
π
180 ◦ ∗ θ
This assumes that the angle has been expressed in degrees to begin with. If an
angle is expressed in radian measure, then to convert it into degrees, simply multiply
by 180◦
π
.