Degrees

A degree is a unit for measuring angles such that 360° make a full circle.

It is not SI unit of angular measurement. The SI unit for the same is Radians
We use a small circle ° to represent degree.
When a more finer measure is needed, degrees are expressed in decimals. e.g. 35.67°



Why only 360° in a circle? Probably because the concept was borrowed from 360 days in year. That way, each degree would correspond to a day!!

Radians

A radian is the SI unit for measuring angles. It is the standard unit of angular measure used in many areas of mathematics.
1 radian is equal to about 57.3°

1 Radian is an angle made when the radius of a circle is wrapped around its circumference.



There is another way of looking at this.

We all have learnt that the circumference of circle is $2πr$ i.e. if we traverse along the circumference of a circle and return to the same point from where we started, we travel $2π$ times the radius ($r$) of the circle or we travel through $2π$ radians.

In other words, if we turn 360°, we go through $2π$ radians.

Radian = $\frac{\text{distance travelled}}{radius}$

If the distance travelled is equal to the radius, we calculate radians as
Radian = $\frac{\text{distance travelled}}{radius}$ = $\frac{r}{r}$ = 1

If the distance travelled is one complete round around the circle (360°), we calculate radians as
Radian = $\frac{\text{distance travelled}}{radius}$ = $\frac{2πr}{r}$ = $2π$

This way we can calculate radians if we know the length of the arc and radius of the circle.

Relation between Degrees and Radians

In most mathematical areas, radians, as a unit of angular measure, is more frequently used than degrees.

Conversion from degrees to radians