The formula above allows us to define radian measure using a circle of any radius r: The radian
measure of an angle θ is s/r, where s is the length of the circular arc that subtends θ in a circle
of radius r (see the Figures below).
EXAMPLE:
◦
(a) Find the length of an arc of a circle with radius 10 m that subtends a central angle of 30
.
(b) A central angle θ in a circle of radius 4 m is subtended by an arc of length 6 m. Find the
measure of θ in radians.
Solution:
π
◦
(a) We know that 30
=
. So the length of the arc is
6
π
5π
s = rθ = (10)
=
m
6
3
(b) By the formula θ = s/r, we have
s
6
3
θ =
=
=
rad
r
4
2
EXAMPLE:
◦
(a) Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15
.
(b) A central angle θ in a circle of radius 9 m is subtended by an arc of length 12 m. Find the
measure of θ in radians.
Solution:
π
◦
(a) We know that 15
=
. So the length of the arc is
12
π
7π
s = rθ = (21)
=
m
12
4
(b) By the formula θ = s/r, we have
s
12
4
θ =
=
=
rad
r
9
3
6