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.

Two radians

In the figure above, 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. 

In general, consider an arc whose length is s on a circle of radius r. The measure of the central angle, θ , that intercepts the arc is θ = (s / r) radians.

Radian measure


An example showing how to find the radian measure

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  θ ?

θ = s / r = 20 inches / 4 inches = 5 

Notice that the unit used in the problem above is inch and it was 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.

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