Higher Engineering Mathematics

B

Geometry and trigonometry

14

The circle and its properties

14.1 Introduction

Acircleis a plain figure enclosed by a curved line,
every point on which is equidistant from a point
within, called thecentre.

14.2 Properties of circles

(i) The distance from the centre to the curve is

called theradius,r, of the circle (seeOPin

Fig. 14.1).

Figure 14.1

(ii) The boundary of a circle is called thecircum-

ference,c.

(iii) Any straight line passing through the centre

and touching the circumference at each end is

called thediameter,d(seeQRin Fig. 14.1).

Thusd=2r.

(iv) The ratio

circumference

diameter

=a constant for any

circle.

This constant is denoted by the Greek let-

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

correct to 5 decimal places.

Hencec/d=πorc=πdorc= 2 πr.

(v) Asemicircleis one half of the whole circle.

(vi) Aquadrantis one quarter of a whole circle.

(vii) Atangentto a circle is a straight line which

meets the circle in one point only and does not

cut the circle when produced.ACin Fig. 14.1

is a tangent to the circle since it touches the

curve at pointBonly. If radiusOBis drawn,

then angleABOis a right angle.

(viii) Asectorof a circle is the part of a circle

between radii (for example, the portionOXY

of Fig. 14.2 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.

Figure 14.2

(ix) Achordof a circle is any straight line which

divides the circle into two parts and is termin-

ated at each end by the circumference.ST,in

Fig. 14.2 is a chord.

(x) Asegmentis the name given to the parts into

which a circle is divided by a chord. If the

segment is less than a semicircle it is called a

minor segment(see shaded area in Fig. 14.2).

If the segment is greater than a semicircle it

is called amajor segment(see the unshaded

area in Fig. 14.2).

(xi) Anarcis a portion of the circumference of a

circle. The distanceSRTin Fig. 14.2 is called

aminor arcand the distanceSX YTis called

amajor arc.

(xii) The angle at the centre of a circle, sub-

tended by an arc, is double the angle at the

circumference subtended by the same arc.

With reference to Fig. 14.3,AngleAOC=

2 ×angleABC.

(xiii) The angle in a semicircle is a right angle (see

angleBQPin Fig. 14.3).

Figure 14.3