Higher Engineering Mathematics, Sixth Edition

The circle and its properties 123

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

an arc, is double the angle at the circumference

subtended by the same arc. With reference to

Fig. 13.3,AngleAOC= 2 ×angleABC.

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

angleBQPin Fig. 13.3).

Q

A

P

C

O

B

Figure 13.3

Problem 1. If the diameter of a circle is 75mm,

find its circumference.

Circumference,c=π×diameter=πd
=π( 75 )=235.6mm.

Problem 2. In Fig. 13.4,ABis a tangent to the

circle atB. If the circle radius is 40mm and

AB=150mm, calculate the lengthAO.

A

B

r

O

Figure 13.4

Atangent toacircleisat right anglestoaradiusdrawn
from the point of contact, i.e.ABO= 90 ◦. Hence, using
Pythagoras’ theorem:

AO^2 =AB^2 +OB^2

AO=

√

(AB^2 +OB^2 )=

√

[( 150 )^2 +( 40 )^2 ]

= 155 .2mm

Now try the following exercise

Exercise 55 Further problemson

propertiesof circles

1. If the radius of a circle is 41.3mm, calculate
   the circumference of the circle.
   [259.5mm]


2. Find the diameter of a circle whose perimeter
   is 149.8cm. [47.68cm]
   3. A crank mechanism is shown in Fig. 13.5,
   whereXYis a tangent to the circle at pointX.If
   the circle radiusOXis 10cm and lengthOYis
   40cm, determine the length of the connecting
   rodXY. [38.73cm]
   X

O 40cm Y

Figure 13.5

4. If the circumference of the earth is 40 000km
   at the equator, calculate its diameter.
   [12730km]


5. Calculate the length of wire in the paper clip
   shown in Fig. 13.6. The dimensions are in
   millimetres. [97.13mm]
   2.5rad

2.5rad

3rad

12

6

32

Figure 13.6

13.3 Radians and degrees

Oneradianis defined as the angle subtended at the

centre of a circle by an arc equal in length to the radius.

s

r

O r



Figure 13.7

With reference to Fig. 13.7,

for arc lengths,

θradians=

s

r