# Radians

Source: https://projectalevel.co.uk/maths/trigonometry/radians

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**Introduction**
Radians, like degrees, are a way of measuring angles.

![](https://content.projectalevel.co.uk/img/maths/rad1.gif)

One radian is equal to the angle formed when the arc opposite the angle is equal to the radius of the circle. So in the above diagram, the angle ø is equal to one radian since the arc AB is the same length as the radius of the circle.

Now, the circumference of the circle is 2pr, where r is the radius of the circle. So the circumference of a circle is 2p larger than its radius. This means that in any circle, there are 2p radians.

Therefore 360º = 2p radians.

Therefore 180º = p radians.

So one radian = 180/p degrees and one degree = p /180 radians.

Therefore to convert a certain number of degrees in to radians, multiply the number of degrees by p /180 (for example, 90º = 90 × p /180 radians = p /2). To convert a certain number of radians into degrees, multiply the number of radians by 180/p .

**Arc Length**
The length of an arc of a circle is equal to rø, where ø is the angle, in radians, subtended by the arc at the centre of the circle. So in the below diagram, s = rø .

![](https://content.projectalevel.co.uk/img/maths/rad2.gif)

**Area of Sector**
The area of a sector of a circle is ½ r² ø, where r is the radius and ø the angle in radians subtended by the arc at the centre of the circle. So in the below diagram, the shaded area is equal to ½ r² ø .

![](https://content.projectalevel.co.uk/img/maths/rad3.gif)