Definition of radian measure

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 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 = 2r / r = 2.

Thus, angle θ measures 2 radians.

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.

Two radians

How to find the radian measure of any angle

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.

Radian measure


A few examples showing how to find the radian measure


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.

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 57.2958 degrees, so the amount of rotation for the central angle from the initial side to the terminal side is 57.2958 degrees.

How learn why 1 radian is equal to 57.2958 degrees, check this lesson about converting radians to degrees

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