Basic Engineering Mathematics, Fifth Edition

The circle 231

is greater than a semicircle it is called amajor

segment(see the un-shaded area in Figure 26.2).

(k) Anarcis a portion of the circumference of a cir-

cle. The distanceSRTin Figure 26.2 is called

aminor arcand the distanceSXYTis called a

major arc.

(l) The angle at the centre of a circle, subtended

by an arc, is double the angle at the circumfer-

ence subtended by the same arc. With reference

to Figure 26.3,

AngleAOC= 2 ×angleABC

Q

A

P

C

O

B

Figure 26.3

(m) Theangleinasemicircleisaright angle(seeangle

BQPin Figure 26.3).

Problem 1. Find the circumference of a circle of

radius 12.0cm

Circumference,c= 2 ×π×radius= 2 πr= 2 π( 12. 0 )

=75.40cm

Problem 2. If the diameter of a circle is 75mm,

find its circumference

Circumference,c=π×diameter=πd=π( 75 )

=235.6mm

Problem 3. Determine the radius of a circular

pond if its perimeter is 112m

Perimeter=circumference,c= 2 πr

Hence,radius of pond,r=

c

2 π

=

112

2 π

=17.83cm

Problem 4. In Figure 26.4,ABis a tangent to the

circle atB. If the circle radius is 40mm and

AB=150mm, calculate the lengthAO

A

B

r

O

Figure 26.4

A tangent to a circle is at right angles to a radius drawn

fromthepointofcontact;i.e.,ABO= 90 ◦.Hence,using

Pythagoras’ theorem,

AO^2 =AB^2 +OB^2

from which, AO=

√

AB^2 +OB^2

=

√

1502 + 402 =155.2mm

Now try the following Practice Exercise

PracticeExercise 101 Properties of a circle

(answers on page 351)

1. Calculate the length of the circumference of
   a circle of radius 7.2cm.


2. If the diameter of a circle is 82.6mm, calcu-
   late the circumference of the circle.


3. Determine the radius of a circle whose cir-
   cumference is 16.52cm.


4. Findthediameter of a circlewhose perimeter
   is 149.8cm.


5. A crank mechanism is shown in Figure 26.5,
   whereXYis a tangent to the circle at point
   X. If the circle radiusOXis 10cm and length
   OY is 40cm, determine the length of the
   connecting rodXY.

X

Y

O 40cm

Figure 26.5

6. If the circumference of the earth is 40000km
   at the equator, calculate its diameter.