Arc Length and Radian Measure Topic Index | Algebra2/Trig Index | Regents Exam Prep Center

(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. The 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:
 RADIANS Arc length of a circle:
 DEGREES 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.
 m

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

Examples:

 1.  Convert 50º to radians.            Answer:     2.  Convert to degrees.          Answer:

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

Answer:

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

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