Radian

Radian is the SI unit for measuring angles. The symbol for denoting radian is rad.

Radian is the unit of angle measurement in the circular system.

Radian is one of the three units of measuring angles.

The other two units of angle measurement are degrees and grades.

Degrees is the unit of angle measurement in the sexagesimal system; and grades, in the centesimal system.

Definition of Radian

Radian is the measure of the angle subtended at the centre of a circle by an arc whose length is equal to radius of the circle.

Think it this way:

If the length of an arc is equal to radius of the circle, then the angle subtended by this arc at the centre will be of what measure?

It is Radian.

In the above circle, AB is an arc whose length l is equal to the radius of the circle, r.
If this arc AB subtends angle θ at the centre of the circle, O, then what measure this angle will be of?

It will be one radian.

So,

1. An arc whose length is equal to the radius of the circle, will subtend one radian at the centre.
2. An arc whose length is twice the radius of the circle, will subtend 2 rad (2 radians) at the centre.
3. An arc will subtend 4rad the centre, if its length is 4 times the radius.

And finally, when the length of the arc is 2Π times the circle radius, then it will subtend 2Π rad the centre.

That is,

If the length of the arc, l =2Π×r and angle subtended is Θ, then
θ = 2Π rad

Now, in a circle, what is the length of the arc whose length is 2Π×r?

It is circumference.

And again, what angle will circumference subtend at the centre of the circle in degrees?

It is 360⁰

So, there we have 2Π rad = 360⁰

Therefore, Π rad = 180⁰

And finally,

1 rad = 180⁰/ Π = 57.295...

Or approximately 57.3⁰