In earlier work, angles were measured in degrees where 1**°** = of one revolution. i.e. 360**°** = one revolution.

This is not a particularly useful way of measuring angles as most formulae for circles use π.

Other ways of measuring angle size are radians(see below) and grads, which will not be studied further.

(1 gradian or grad = 1/100 of a right angle or 0.9 of a degree.)

Students should be familiar with both degrees and radians as a means of measuring angles.

Make sure that calculators are set to the mode of angle measurement required.

### Radians

A radian is defined as the angle made at the centre of a circle, of radius r, by an arc of length r.

e.g.

This would be true for every circle regardless of size.

The circumference of a circle = 2π r therefore an angle of 360**° = **2π r ÷ r = **2π radians**

Changing Degrees to Radians

360° =2π radians 1° =radians =radians

To change DegreestoRadiansmultiply by

Changing Radians to Degrees

2π radians =360 ° 1 radian =° =°

To change RadianstoDegreesmultiply by

DegreesRadians0 030 π/645 π/490 π/2135 3π/4150 5π/6180 π225 5π/4270 3π/2315 7π/4360 2π

### Length of an Arc

The length of an arc of a circle can now be calculated. An arc is said to be **minor **if the angle **subtended** at the centre is less than 180**°.**

### Area of a Sector

The area of a sector of a circle can also be found. A sector is said to be **minor **if the angle **subtended** at the centre is less than 180**°**

Example of Length of Arc |
Example of Area of Sector |

Find the length of arc Angle is already in radians. Length of arc = r θ = 7.8 x 7π/8 = |
Find the area of sector AOB. First change angle from degrees to radians. 125 Area of sector AOB = 0.5 x r = 0.5 x 6 = |