The circle 231
is greater than a semicircle it is called a major
segment (see the un-shaded area in Figure 26.2).
(k) An arc is a portion of the circumference of a circle. The distance SRT in Figure 26.2 is called
a minor arc and the distance SXYT is called a
major arc.
(l) 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 Figure 26.3,
Angle AOC = 2 × angle ABC
Q
A
P
C
O
B
Figure 26.3
(m) The angle in a semicircle is a right angle (see angle
BQP in Figure 26.3).
Problem 1. Find the circumference of a circle of
radius 12.0 cm
Circumference, c = 2 × π × radius = 2πr = 2π(12.0)
= 75.40 cm
Problem 2. If the diameter of a circle is 75 mm,
find its circumference
Circumference, c = π × diameter = πd = π(75)
= 235.6 mm
Problem 3. Determine the radius of a circular
pond if its perimeter is 112 m
Perimeter = circumference, c = 2πr
Hence, radius of pond, r =
c
2π
=
112
2π
= 17.83 cm
Problem 4. In Figure 26.4, 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 26.4
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
from which,
AO =
AB 2 + OB 2
=
150 2 + 40 2 = 155.2 mm
Now try the following Practice Exercise
Practice Exercise 101 Properties of a circle
(answers on page 351)
1. Calculate the length of the circumference of
a circle of radius 7.2 cm.
2. If the diameter of a circle is 82.6 mm, calculate the circumference of the circle.
3. Determine the radius of a circle whose circumference is 16.52 cm.
4. Find the diameter of a circle whose perimeter
is 149.8 cm.
5. A crank mechanism is shown in Figure 26.5,
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 .
X
Y
O
40 cm
Figure 26.5
6. If the circumference of the earth is 40 000 km
at the equator, calculate its diameter.
is greater than a semicircle it is called a major
segment (see the un-shaded area in Figure 26.2).
(k) An arc is a portion of the circumference of a circle. The distance SRT in Figure 26.2 is called
a minor arc and the distance SXYT is called a
major arc.
(l) 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 Figure 26.3,
Angle AOC = 2 × angle ABC
Q
A
P
C
O
B
Figure 26.3
(m) The angle in a semicircle is a right angle (see angle
BQP in Figure 26.3).
Problem 1. Find the circumference of a circle of
radius 12.0 cm
Circumference, c = 2 × π × radius = 2πr = 2π(12.0)
= 75.40 cm
Problem 2. If the diameter of a circle is 75 mm,
find its circumference
Circumference, c = π × diameter = πd = π(75)
= 235.6 mm
Problem 3. Determine the radius of a circular
pond if its perimeter is 112 m
Perimeter = circumference, c = 2πr
Hence, radius of pond, r =
c
2π
=
112
2π
= 17.83 cm
Problem 4. In Figure 26.4, 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 26.4
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
from which,
AO =
AB 2 + OB 2
=
150 2 + 40 2 = 155.2 mm
Now try the following Practice Exercise
Practice Exercise 101 Properties of a circle
(answers on page 351)
1. Calculate the length of the circumference of
a circle of radius 7.2 cm.
2. If the diameter of a circle is 82.6 mm, calculate the circumference of the circle.
3. Determine the radius of a circle whose circumference is 16.52 cm.
4. Find the diameter of a circle whose perimeter
is 149.8 cm.
5. A crank mechanism is shown in Figure 26.5,
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 .
X
Y
O
40 cm
Figure 26.5
6. If the circumference of the earth is 40 000 km
at the equator, calculate its diameter.