1. Radian
Measure

2. Trigonometry: Radians
KUS objectives
BAT convert fluently between degrees and radians
BAT calculate arcs, sectors and segments
Starter: Simplify (without a calculator)
sin cos 60
2 sin 45 + cos
tan − 3 tan 120

3. Introduction to Radian measure1
For a circle of radius r, the angle at the centre, which subtends an arc of length r, is equal to 1 radian
‘If arc AB has length r, then angle AOB is 1 radian (1c or 1 rad)’
As the circumference of the circle is 2πr the total angle is 2π radians
Radians are an alternative to degrees. Some calculations involving circles are easier when Radians are used, as opposed to degrees. r A B O 1c
45  
0    120  180 210
𝜋 𝑐
0 𝑐

4. WB 1 Convert the following angles to degrees
Converting radians and degrees
Radians  Degrees 𝜋 𝑐 =180
WB 1 Convert the following angles to degrees
7𝜋 8 𝑐
4𝜋 15 𝑐 a)
1.8 𝑐 b) c)
=1.8 × 180 𝜋
= 7𝜋 8 × 180 𝜋
= 4𝜋 15 × 180 𝜋
=103°
= 1260𝜋 8𝜋 =157.5°
= 720𝜋 15𝜋 =48°

5. Now do skills 249 35 0 =110× 𝜋 180 =150× 𝜋 180 =35× 𝜋 180 = 5𝜋 6 𝑐
Converting degrees to radians
Degrees  Radians 𝜋 𝑐 =180
WB Convert the following angles to radians d) e) f)
35 0
=150× 𝜋 180
=110× 𝜋 180
=35× 𝜋 180
= 5𝜋 6 𝑐
= 11𝜋 18
= 7𝜋 36 = 𝑐
The superscript 𝑐 is often unwritten
Now do skills 249

6. ARC Length

7. This is the formula’s usual form
SECTOR area
The Area of a Sector and Segment can be worked out using Radians = A =
Multiply by π O θ X =
Multiply by r2 B = =
This is the formula’s usual form

8. r= 10 𝜃 + 2 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟=𝑟𝜃+2𝑟 10=𝑟(𝜃+2) 10 𝜃+2 =𝑟
WB 2a arc length
Arc AB of a circle, with centre O and radius r, subtends an angle of θ radians at O. The Perimeter of sector AOB is 10 cm. Express r in terms of θ.
Length AB = rθ
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟=𝑟𝜃+2𝑟
10=𝑟(𝜃+2)
10 𝜃+2 =𝑟 r θ A B O rθ
r= 10 𝜃 + 2

9. Finding the length of an arc is easier when you use radians
WB 3 arc length
Calculator in Radians
Finding the length of an arc is easier when you use radians
The border of a garden pond consists of a straight edge AB of length 2.4m, and a curved part C, as shown in the diagram below. The curved part is an arc of a circle, centre O and radius 2m. Find the length of C. O
(H) 2m x A
1.2m B
Inverse sine
(O)
Double for angle AOB O A B
2.4m 2m C θ
Angle θ = 2π – 1.287
Angle θ = rad
(We need to work out angle θ)

10. WB 4 sector area
In the diagram, the area of the minor sector AOB is 28.9cm2. Given that angle AOB is 0.8 rad, calculate the value of r.
Put the numbers in
½ x 0.8 = 0.4 B O
0.8c A
r cm
Divide by 0.4
Square root

11. WB 5 sector area
A plot of land is in the shape of a sector of a circle of radius 55m. The length of fencing that is needed to enclose the land is 176m. Calculate the area of the plot of land.
The length of the arc must be 66m (adds up to 176 total)
Put the numbers in
Divide by 55 A
55m
66m
1.2c θ O
Put the numbers in
55m B
(We need to work out the angle first)
Now do skills 251

12. It is not necessary too memorise this formula – use common sense
Segment area
Work out the area of a segment using radians.
Area of a Segment
Area of Sector AOB – Area of Triangle AOB
Area of Sector AOB
Area of Triangle AOB O r r θ
a = b = r
C = θ A B
Area of the Segment
Factorise
It is not necessary too memorise this formula – use common sense

13. Calculate the Area of the segment shown in the diagram below.
WB 6 segment area
Calculate the Area of the segment shown in the diagram below.
Substitute the numbers in
Work the parts out O
π 3
2.5cm
Only round the final answer

14. Remember, sin x = sin (180 – x)
WB 7 segment area
In the diagram AB is the diameter of a circle of radius r cm, and angle BOC = θ radians. Given that the Area of triangle AOC is 3 times that of the shaded segment, show that 3θ – 4sinθ = 0
Area of the shaded segment
Area of triangle AOC
a = b = r Angle = π-θ
Remember, sin x = sin (180 – x) A C B θ
AOC = 3 x shaded segment
Cancel out 1/2r2
Multiply out the brackets
Subtract sinθ

15. WB 8
Figure 2 shows a plan of a patio. The patio PQRS is in the shape of a sector of a circle with centre Q and radius 6 m.
Given that the length of the straight line PR is 63 m,
(a) find the exact size of angle PQR in radians.
(b) Show that the area of the patio PQRS is 12 m2.
(c) Find the exact area of the triangle PQR.
(d) Find, in m2 to 1 decimal place, the area of the
segment PRS.
(e) Find, in m to 1 decimal place, the perimeter of the
patio PQRS.

16. WB 9

17. WB 9 exam Q - solution

18. WB 10

19. One thing to improve is –
KUS objectives BAT convert fluently between degrees and radians
BAT calculate arcs, sectors and segments
self-assess
One thing learned is –
One thing to improve is –

20. END