(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
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.