- 1 Revolution = 360°
- radian = 180°
- 1 radian = = 57.3°
- 1° = radian = 0.175 radian
- Length of arc
- Area of a Sector
- Area of a segment
The most common system of measuring the angles is that of degrees. One complete revolution is divided into 360 equal parts and each part is called one degree (1°). Furthermore, Half revolution is equivalent to °.
Also for smaller angles, 1° is divided into 60 parts and each part is called 1 minute which is further divided into 60 parts and each part is now called 1 second.
1 Revolution = 360°
1 Degree = 60 minutes
1 minute = 60 seconds
What are Radians?
Radians are a very important way of measuring angles. A radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to that of the radius of the circle.
The diagram ‘Fig 1’ shows a circle with centre 0 and radius r. The length of the arc AB is also r. In this position, the angle made by the arc AB at the centre is defined as 1 radian.
If AB = r, then radian where is the angle subtended at the center.
If AB = 2r then radian
If which is the circumference then is radian. Shown in Fig 2.
As is the circumference , then OA makes a complete revolution °
radian = 180°
1 radian = = 57.3°
1° = radian = 0.175 radian
To convert an angle from degrees to radians we multiply the angle by radian e.g:
Q. Convert 150° angle into radian.
Length of Arc
Let s be length of an arc AB which subtends an angle 𝛳 radian at the centre O of a circle of radius r units. Shown in Fig 3.
From the definition of radian, we calculate that:
where is in radians
Area of a sector
To find the area of the sector which contains angle 𝛳 radian at the centre of the circle
as shown in Fig 4, we consider the sector as a fraction of the circle hence:
Area of a segment
The area of the minor segment as shown in Fig 5
is obtained by the arc AB of the centre O is given by:
where is in radians
Q. The diagram (Fig 6) shows the sector OPQ of a circle with centre O and radius r cm.
The angle POQ is radians and the perimeter of the sector is 20 cm.
i) Show that
ii) Find the area of the sector in terms of r.
i) Perimeter = 20
OP + OQ + PQ = 20
ii) Area of the sector
Substituting from part (i) above
- Edexcel AS and A level Modular Mathematics C2