**Angular displacement** is often expressed in one of three units. The first is the degree, and it is well known that there are 360 degrees in a circle.

The second unit is the revolution (rev), one revolution representing one complete turn of 360 degrees.

The most important unit from a scientific viewpoint, however, is the **SI unit called the radian (rad)**. Here we will quickly find out how to convert between degrees and radians.

**How to convert between degrees and radians? | degree and radian conversion**

We will take example of a CD (Compact Disc). Figure 1 shows how the radian is defined. A point P on the disc starts out on the stationary reference line, so that **θ _{0}** =0 rad, and the angular displacement is ∆

**θ = θ**–

**θ**.

_{0}

Now along with angular displacement, the CD makes linear displacement as well. As the disc rotates, the point traces out an arc of length s, which is measured along a circle of radius r. When **θ** is very minute, arc s is small straight line

Hence, Sin **θ** = s/r ….. (1)

As **θ** is minute, we can take Sin **θ** = **θ**.

So, modifying equation (1) we get, **θ** = s/r …………(2)

Equation (2.i) now defines the angle **θ** in radians:

**θ (in radians) = Arc Length / Radius = s/r** …………….(2.i)

According to this definition, an angle in radians is the ratio of two lengths; for example,

meters/meters. In calculations, therefore, the radian is treated as a number without units.

To convert between degrees and radians, it is only necessary to remember that the arc

length s of an entire circle of radius r is the circumference 2πr.

Therefore, according to Equation 2, the number of radians that corresponds to 360 degrees, or one revolution, is **θ** = s/r=2πr/r = 2π rad**So, 360 degrees = 2π rad ……………………….. (3)**

Again,

**2π rad**= 360 degrees

**So, 1 rad = 360 degrees/2π = 57.3 degrees………………..(4)**

So, we get two equations (3) and (4) to do the convertion between radian and degree.