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There is another system of angular measurement, called the Circular System. It is most useful for the study of higher mathematics. Specially in Calculus, angles are measured in radians.
Radian:
Radian is the measure of the angle subtended at the center of the circle by an arc, whose length is equal to the radius of the circle.
Consider a circle of radius . Construct an angle  at the center of the circle whose rays cut off an arc  on the circle whose length is equal to the radius  .
Thus  radian
Relation between the length of an arc of a circle and the circular measure of its central angle:
Prove that 
Where  is the radius of the circle  , is the length of the arc and  is the circular measure of the central angle.
Proof:
Let there be a circle with center  and radius  . Suppose that length of arc  and the central angle  radian. Take an arc  of length  .
By definition  radian.
We know from elementary geometry that measures of central angles of the arcs of a circle are proportional to the lengths of their arcs.
 
 
Thus the central angle  (in radian) subtended by a circular arc of length  is given by  , where  is the radius of the circle.
Remember that  and  are measured in terms of the same unit and the radian measure is unit-less, i.e., it is a real number.
For example, if  and 
then 
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