Arc Length and Radian Measure
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(The arc length discussed on this page will be in relation to a circle.)

An arc of a circle is a "portion" of the circumference of the circle.

length of an arc is simply the length of its "portion" of the circumference.  Actually, the circumference itself can be considered an arc length.

The length of an arc (or arc length) is traditionally symbolized by s.

In the diagram at the right, it can be said that " subtends angle ".
Definition:  subtend - to be opposite to
The radian measure of a central angle of a circle is defined as the ratio of the length of the arc the angle subtends, s, divided by the radius of the circle, r.

From this definition we can obtain:

Arc length of a circle:

Arc length of a circle:

When working in the unit circle, with radius 1, the length of the arc equals the radian measure of the angle.

= 1 radian

A radian is the measure of an angle   that , when drawn as a central angle, subtends an arc whose length equals the length of the radius of the circle.

Relationship between Degrees and Radians:

When the arc length equals an entire circumference, we can use to get
  This implies that .

So,    and 

To change
 from degrees to radians,
multiply by

To change
 from radians to degrees,
multiply by


1.  Convert 50 to radians.


2.  Convert to degrees.


3.   How long is the arc subtended by an angle of radians on a circle of radius 20 cm?



How to use your
TI-83+/84+ graphing calculator  with radians and degrees.
Click calculator.