1. RADIANS
Definition
An arc of length r subtends an angle of one radian at the centre of a circle of radius r r
Ø=1radian

2. Proof r How do you calculate the length of an arc? r = ø x 2πr 360º
Ø=1radian
How do you calculate the length of an arc?
r = ø x 2πr
360º
r = 1 radian x 2πr
360 º
x 360 º
360r = 1 radian x 2πr
÷ r
360 º = 1 radian x 2π
÷ 2
180 º = 1 radian x π
Or 180 º = π radians
÷ π
180 = 1 radian π
so 1 radian is approximately…?

3. Converting between angles and radians
Degrees = radians x 180 π
Radians = degrees x π
180
So if ø is measured in radians
Then ø radians = ø x π
180

4. How many different angles can you write as radians?2
180º = π radians

5. Arc Length r Angle in degrees Arc length = ø x 2πr 360º
Ø=1radian
Angle in degrees
Arc length = ø x 2πr
360º
Arc length = 2πrø
360º
Factorise r
Arc length = r 2πø
360º
Divide by 2
Angle in radians
Arc length = r πø
180º
Arc length = rø

6. Area of Sector r Angle in degrees Sector area = ø x πr2 360º
Ø=1radian
Angle in degrees
Sector area = ø x πr2
360º
Sector area = πr2ø
360º
Factorise r2
Sector area = r2 πø
360º
Factorise out ½
Angle in radians
Sector area = ½r2 πø
180º
Sector area = ½r2ø

7. Examples
Convert 50° into radians
50° = ° x π rad
180
50° = rad

8. Examples Convert 2.7 radians into degrees 2.7 rad = 2.7 x 180 degreesπ
2.7 rad = °

9. Examples Convert 40° into radians 40° = 40 x π 180 40° = 40π 180
40° = 40π
180
40° = 2π radians 9

10. Calculate the arc length and sector area
10cm
1.2radians
Arc length = rө
Arc length = 10 x 1.2
Arc length = 12cm
Sector area = ½r2ө
Area = ½ x 100 x 1.2
Area = 60cm2