rotation.jpgIn earlier work, angles were measured in degrees where 1° = Y11_Radians_Arcs_and_Sectors_01.gif of one revolution. i.e. 360° = one revolution.

This is not a particularly useful way of measuring angles as most formulae for circles use π.

Other ways of measuring angle size are radians(see below) and grads, which will not be studied further. 
(1 gradian or grad = 1/100 of a right angle or 0.9 of a degree.)

Students should be familiar with both degrees and radians as a means of measuring angles. 
Make sure that calculators are set to the mode of angle measurement required.

Radians

A radian is defined as the angle made at the centre of a circle, of radius r, by an arc of length r.

e.g. Y11_Radians_Arcs_and_Sectors_02.gif

This would be true for every circle regardless of size.

The circumference of a circle = 2π r therefore an angle of 360° = 2π r ÷ r = 2π radians

 

Changing Degrees to Radians

360°
=
2π radians
1°
=
Y11_Radians_Arcs_and_Sectors_03.gif radians
 
=
Y11_Radians_Arcs_and_Sectors_05.gifradians

 

To change Degrees to Radians multiply by Y11_Radians_Arcs_and_Sectors_04.gif

Changing Radians to Degrees

2π radians
=
360°
1 radian
=
°
 
=
Y11_Radians_Arcs_and_Sectors_07.gif°

 

To change Radians to Degrees multiply by Y11_Radians_Arcs_and_Sectors_08.gif
Degrees
Radians
0
0
30
π/6
45
π/4
90
π/2
135
3π/4
150
5π/6
180
π
225
5π/4
270
3π/2
315
7π/4
360

 

 

Length of an Arc

The length of an arc of a circle can now be calculated. An arc is said to be minor if the angle subtended at the centre is less than 180°.

Y11_Radians_Arcs_and_Sectors_09.gif

Area of a Sector

The area of a sector of a circle can also be found. A sector is said to be minor if the angle subtended at the centre is less than 180°

Y11_Radians_Arcs_and_Sectors_10.gif

 

Example of Length of Arc
Example of Area of Sector

Find the length of arc AB

Y11_Radians_Arcs_and_Sectors_11.gif

Angle is already in radians.

Length of arc = r θ

                      = 7.8 x 7π/8

                      = 21.4 mm (to 3 sig.fig.)

Find the area of sector AOB.

Y11_Radians_Arcs_and_Sectors_12.gif

First change angle from degrees to radians.

125° = 125 x Y11_Radians_Arcs_and_Sectors_13.gif = 2.18 radians (to 3 sig. fig.)

Area of sector AOB = 0.5 x r 2 θ

                                 = 0.5 x 62 x 2.18

                                 = 39.2 cm(to 3 sig. fig.)