Chapter 26

The circle

26.1 Introduction

Acircleis aplain figureenclosed by acurved line,every
point on which is equidistantfrom a pointwithin, called
thecentre.
InChapter25,workedproblemsontheareasofcircles
and sectors were demonstrated. In this chapter, proper-
ties of circles are listed and arc lengths are calculated,
together with more practical examples on the areas of
sectors of circles. Finally, the equation of a circle is
explained.

26.2 Properties of circles

(a) The distance from the centre to the curve is called

theradius,r,ofthecircle(seeOPinFigure26.1).

C

B

Q

O

P

R

A

Figure 26.1

(b) The boundary of a circle is called thecircumfer-

ence,c.

(c) Any straight line passing through the centre and

touching the circumference at each end is called

thediameter,d(seeQRin Figure 26.1). Thus,

d= 2 r.

(d) The ratio

circumference

diameter

is a constant for any cir-

cle. This constant is denoted by theGreek letterπ

(pronounced ‘pie’), whereπ= 3 .14159, correct

to 5 decimal places (check withyour calculator).

Hence,

c

d

=πorc=πdorc= 2 πr.

(e) Asemicircleis one half of a whole circle.

(f) Aquadrantis one quarter of a whole circle.

(g) Atangenttoa circleis astraightlinewhich meets

thecircleatonepointonlyanddoesnotcutthecir-

cle when produced.ACinFigure 26.1 isa tangent

to the circle since it touches the curve at pointB

only. If radiusOBis drawn,angleABOis a right

angle.

(h) The sectorof a circle is the part of a circle

between radii (for example, the portionOXYof

Figure 26.2 is a sector). If a sector is less than a

semicircle it is called aminor sector; if greater

than a semicircle it is called amajor sector.

X

Y

S T

R

O

Figure 26.2

(i) Thechordof a circle is any straight line which

divides the circle into two parts and is termi-

nated at each end by the circumference.ST,in

Figure 26.2, is a chord.

(j) Segmentis thename given to theparts intowhich

a circleis dividedby achord.If thesegment is less

than a semicircle it is called aminor segment

(see shaded area in Figure 26.2). If the segment

DOI: 10.1016/B978-1-85617-697-2.00026-0