Cambridge Additional Mathematics

The unit circle and radian measure (Chapter 8) 205

The diagram alongside illustrates terms relating to the parts

of a circle.

An arc, sector, or segment is described as:

² minorif it involves less than half the circle

² majorif it involves more than half the circle.

For example:

ARC LENGTH

Forμinradians, arc length l=μr.

Forμindegrees, arc length l= 360 μ £ 2 ¼r.

AREA OF SECTOR

Forμinradians, area of sector A=^12 μr^2.

Forμindegrees, area of sector A= 360 μ £¼r^2.

B Arc length and sector area

minor segment

major segment

A

B major arc AB

(red)

minor arc AB

(black)

segment

sector

radius

arc (part of circle)

centre

chord

O

A

B

l

q r

O

X

Y

q r

r

Radians are used in pure

mathematics because they

make formulae simpler.

In the diagram, thearc lengthAB isl.

Angleμis measured inradians.

We use a ratio to obtain:

arc length

circumference

=

μ

2 ¼

)

l

2 ¼r

=

μ

2 ¼

) l=μr

In the diagram, the area of minor sector XOY is shaded.

μis measured inradians.

We use a ratio to obtain:

area of sector

area of circle

=

μ

2 ¼

)

A

¼r^2

=

μ

2 ¼

) A=^12 μr^2

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_08\205CamAdd_08.cdr Monday, 6 January 2014 9:35:40 AM BRIAN