Different units are often used to measure the same entity. The height of a person is expressed using meters, feet or inches. Distance between places is expressed using kilometers, miles or even light-years.
Similarly, we often use degree and radian to measure angles.
In this chapter, we shall explore radian as a unit to measure an angle. Also, we shall see how radian and degree are related to each other.
Let's say \(\angle{AOB} = \theta\) is the central angle (an angle made at the centre of the circle) while moving along the circumference from point A to B.
The measure of θ in radians is the ratio of the length of the arc AB (arc AB = s) to the radius (radius = r).
θ = Arc Length ∕ Radius Length = s ∕ r radian. Being a ratio of two same units of length, radian has no unit.
Example 1: Length of AB = 8 cm and radius r = 4 cm, then \(\theta = \frac{8}{4}=2 \textrm{ radians}\).
Example 2: Length of AB = π ∕ 4 cm and radius be 1 cm, then θ = π ∕ 4 radian
Example 3: Now consider the whole circumference of the circle which is 2πr, where r = radius of the circle.
The central angle that the circumference makes = \(\frac{2 \pi r }{r}\) = 2π radians.
As going around the whole circle makes an angle of \(360^{\circ}\) at the centre.
Therefore, 2π radians = 360° ⇒ π radians = 180°.
The following table shows the conversion between radian and degree.
| Degree | Radian |
| 360° | 2π |
| 180° | π |
| 90° | π ∕ 2 |
| 45° | π ∕ 4 |
