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
- If the radius of a circle is 41.3mm, calculate
the circumference of the circle.
[259.5mm] - 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
- If the circumference of the earth is 40 000km
at the equator, calculate its diameter.
[12730km] - 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