**Summary**

- 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

#### Example #1

To convert an angle from degrees to radians we multiply the angle by radian e.g:

Q. Convert 150° angle into radian.

*Solution:*

150°= 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

#### Example #2

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*.

*Solution:*

i) Perimeter = 20

OP + OQ + PQ = 20

Hence shown:

ii) Area of the sector

Substituting from part (i) above

##### Reference

- Edexcel AS and A level Modular Mathematics C2