FURTHER MATHEMATICS
SS 2
Topic:
Circular Measure
Content:
Radians
Length of an Arc
Area of Sector
Perimeter of a sector
Procedure:
Step 1:
Radians
Angles can be measured in degrees or radians.
A radian is a larger unit which is used in trigonometry.
It is defined as the angle subtended at the centre of a circle by an arc equal to the
radius of the circle.
P
0 1 radian
arc PQ
= 1 unit
1 unit
Q
Step II:
Relationship between radian and degree
1 revolution = 3600 = 2π radians
2π radians = 3600
1 radian = 3600 = 1800
2π
π
3600 = 2π radian
10
= 2π radian
360
10
= π radian
180
Example 1: Convert to degrees
(i) π rad
4
Solutions
(i)
π x 180 =
4x π
(ii)
(ii) 7π rad
2
1800 = 450
4
7π x 180 = 7 x 900 = 6300
2 x π
Example 2: Convert to radians, leaving π in your answer
(i) 1500
(ii) 4500
Solutions:
(i)
1500 x π/180 = 5/6 π radian
(ii)
4500 x π/180 = 15/6 π = 5/2π radian
Step III: Length of an Arc (radian)
Q
S
r
O
Arc PQ = θr
Area of sector (radians)
Area of sector = θ
Area of circle
2π
Area of sector = θ
πr2
2π
Area of sector = ½ r2θ
Perimeter of the sector
= r + r + rθ
P
θ
r
Perimeter = 2r + rθ
Step IV: Worked examples
Example 3: Find the length of an arc which subtends an angle of 0.8 radians at the
centre of a circle of radius 10cm.
Solution:
θ = 0.8rad
r = 10cm
Arc = rθ = 0.8 x 10 = 8cm
Example 4: Find the angle subtended at the centre of a circle of radius 2.5cm by an
arc 2cm long.
Solution
arc = rθ
2 = 2.5θ
θ = 2/2.5 = 0.8 radian
Example 5: An arc subtends an angle of 1 radian at the centre of a circle, and a
sector of area 72cm2 is bounded by this arc and the two radii. Find the radius of the
circle.
Solution:
θ
=
1 radian
Area =
72cm2
Area =
½ θ r2
72 = ½ x 1/1 x r2
cross multiply
72 x 2 = r2
144 = r2
r = √144 = 12cm
Summary
(i)
1 radian = 180/π 0
(ii)
1 degree = π/180 0
(iii) Length of an Arc = rθ
(iv) Area of a sector = ½ r2θ
(v)
Perimeter of a sector = 2r + rθ
ASSIGNMENT
1.
Express
(a) 700
(b) 2500 in radians correct to two decimal places
2.
Express
(a) 3rad
(b) 2.5rad in degrees correct to the nearest degree
3.
4.
Find the angle subtended at the centre by an arc of length 3cm on a circle of radius
3cm.
Find the area of a sector of a circle of radius 8cm if the arc of the sector subtends
5.
an angle of 0.25 radians at the centre.
A disc makes 100 revolutions in three minutes. Find the angle through which it
turns every second, in radians and in degrees.
Direct all your enquiries to Tony Nwaneri (07033806316)
Submit work done, online, to [email protected].