You already know that angles are measured by degrees (360⁰ is a complete revolution). An alternative method is based on the circumference of a circle.

If an arc of a circle is drawn such that it is the same length as the radius, then the angle created is called one radian (1^C), as shown below.

From the diagram you can see that dividing the circumference by the radius will give the number of radians in one complete revolution. Therefore, the number of radians in one revolution is,

These basic conversions are useful to know:

Fractions of 180⁰ can be written in radians by using 180⁰ = π rad

For example:

You also need to be able to convert an angle given in radians to degrees:

For example:

This means that one radian is just a little less than 60⁰.

Arc length

In the diagram below we can see that for a given angle θ, the length of the arc is rθ. (See if you can calculate this using the fact that the arc is θ/2π of the whole circumference.)

Area of a sector

Using the same idea, we find that the area of a sector of a circle is ½ r²θ.

(Again, see if you can derive this formula using the same argument as above.)