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, Angle AOC = 2 × angle ABC.
(xiii) The angle in a semicircle is a right angle (see
angle BQP in Fig. 13.3).
Q
A
P
C
O
B
Figure 13.3
Problem 1. If the diameter of a circle is 75 mm,
find its circumference.
Circumference, c = π × diameter = πd
= π(75) = 235.6 mm.
Problem 2. In Fig. 13.4, AB is a tangent to the
circle at B. If the circle radius is 40 mm and
AB = 150 mm, calculate the length AO.
A
B
r
O
Figure 13.4
A tangent to a circle is at right angles to a radius drawn
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 ]
= 155.2 mm
Now try the following exercise
Exercise 55 Further problems on
properties of circles
1. If the radius of a circle is 41.3 mm, calculate
the circumference of the circle.
[259.5 mm]
2. Find the diameter of a circle whose perimeter
is 149.8 cm.
[47.68 cm]
3. A crank mechanism is shown in Fig. 13.5,
where XY is a tangent to the circle at point X. If
the circle radius OX is 10 cm and length OY is
40 cm, determine the length of the connecting
rod XY.
[38.73 cm]
X
Y
O
40 cm
Figure 13.5
4. If the circumference of the earth is 40 000 km
at the equator, calculate its diameter.
[12 730 km]
5. Calculate the length of wire in the paper clip
shown in Fig. 13.6. The dimensions are in
millimetres.
[97.13 mm]
2.5 rad
2.5 rad
3 rad
12
6
32
Figure 13.6
13.3 Radians and degrees
One radian is 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 length s,
θ radians =
s
r