Arc Length and Radian Measure

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.

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<COD = 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