1. C2:Radian Measure
Learning Objective: to understand that angles can be measured in radians

2. Measuring angles in degrees
An angle is a measure of rotation.
The system of using degrees to measure angles, where 1° is equal to of a full turn, is attributed to the ancient Babylonians.
The use of the number 360 is thought to originate from the approximate number of days in a year.
360 is also a number that has a high number of factors and so many fractions of a full turn can be written as a whole number of degrees.
Another unit of rotation is the “gradian”, where 1 gradian is 1/400 of a full turn and so a right angle is 100 gradians. Most calculators have a ‘grad’ option although this unit is rarely used.
For example, of a full turn is equal to 160°.

3. Measuring angles in radians
In many mathematical and scientific applications, particularly in calculus, it is more appropriate to measure angles in radians.
A full turn is divided into 2π radians.
Remember that the circumference of a circle of radius r is equal to 2πr.
One radian is therefore equal to the angle subtended by an arc of length r. r
1 rad O
1 radian can be written as 1 rad or 1c.
2π rad = 360°
So: rad =

4. Converting radians to degrees
We can convert radians to degrees using:
2π rad = 360°
Or:
π rad = 180°
Radians are usually expressed as fractions or multiples of π so, for example:
If the angle is not given in terms of π, when using a calculator for example, it can be converted to degrees by multiplying by
For example:

5. Converting degrees to radians
To convert degrees to radians we multiply by
For example: 10 3 9
Sometimes angles are required to a given number of decimal places, rather than as multiples of π, for example:
Note that when radians are written in terms of π the units rad or c are not normally needed.

6. Important angles to learn:

7. Converting between degrees and radians

8. Task 1:
Exercise 6A
Radians to Degrees : multiply by 360, divide by 2π.
Degrees to Radians : multiply by 2π, divide by 360.