The radian measure of any central angle is the length of the intercepted arc divided by the length of the radius of the circle.
In other words, it is the ratio of the arc length to the length of the radius of the circle.
In the figure below, notice that the central angle of the circle intercepts an arc whose length is twice the length of the radius of the circle.
The radian measure of the angle θ = length of the intercepted arc / length of radius
θ = 2r / r = 2. Thus, angle θ measures 2 radians or simply 2 rad.
By the same token, the measure of a central angle that intercepts an arc whose length is equal to the length of the radius of the circle is 1 radian since r / r = 1.
In general, consider an arc whose length is s on a circle of radius r. The radian measure of the central angle, θ , that intercepts the arc is θ = (s / r) radians.
If the angle θ makes a full rotation, then the length of the arc is the circumference of the circle.
The formula to use to find the circumference of a circle is 2pir.
s = 2pir
θ = (2pir / r) radians.
θ = 2pi radians or 2π radians with pi = π = 3.14
Example #1:
A circle has a radius equal to 4 inches. If a central angle, θ , intercepts an arc of length 20 inches, what is the radian measure of the central angle θ ?
θ = s / r = 20 inches / 4 inches = 5
Example #2:
A circle has a radius equal to 9 centimeters. If a central angle, θ , intercepts an arc of length 36 centimeters, what is the radian measure of the central angle θ ?
θ = s / r = 36 centimeters / 9 centimeters = 4
Notice that the units used in the problems above are inches and centimeters and they were canceled when we use the formula for radian measure. Now, we are left with a number that does not have a unit. Therefore, if an angle has a measure of 5 radians, we can write θ = 5 radians or simply 5. If the angle is measured in degree, then we must use the degree symbol. If the degree symbol is not used, then we can assume that the angle is measured in radians.
Example #3:
A youngster sitting on a merry-go-round moves clockwise through an arc length of 2.355 meters. If the youngster’s angular displacement is 1.57 rad, how far is the youngster sitting from the center of the merry-go-round?
What we are looking for is the radius of the merry-go-round.
Using the formula θ = s / r, we know that r = s / θ
r = 2.355 / 1.57
r = 1.5 m
Since the radius of the merry-go-round is 1.5 m, the distance the youngster is sitting from the center is 1.5 m
Just like degrees, radians measure the amount of rotation from the initial side to the terminal side of an angle.
For example, 1 radian is equal to 57.297 degrees, so the amount of rotation for the central angle from the initial side to the terminal side is 57.297 degrees.