Page 125 - Mathematics_Form_3

Circles

5. Find the length of an arc of a circle The expression is called the radian
l _
r
of radius10 cm if the central angle and is denoted by s.
is: Therefore, if s is a radian measure of
(a) 45° (b) 90° (c) 54° an angle and is its degree measure,
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(Leave your answer in terms of π.) then
6. The soccer ball moves in a 6.39 s = ____
πθ
.
metres arc during a corner kick. 180°
Determine the radius of the arc if For instance, for an angle of 270° , then
the ball sweeps at a 40° angle. its radian measure is given by,
7. Show that the radius of a circle with 270°π
an arc of length cm and central s = _____
180°
angle 30° is 6 cm.
3π
=
___
8. What is the length of an arc of 2
3π
the circle of radius 40 cm which Therefore, 270° is equivalent to
___
2
subtends an angle of 25° at the radians. Similarly, radians can be
π _
2
centre? converted into degrees using the formula:
9. Find the length of an arc of a circle 180° s

with radius 32 cm which subtends θ = _____
π
an angle of 155° at the centre of Substituting for s gives,
π _
the circle. Leave your answer in 2
π __
terms of π. 180° ×
2
θ = _______

10. An arc of length 30 m is formed π
from a circle of radius 50 m. What = 90°
angle does the arc subtends? Therefore, radians is equivalent to
π _
2
Mathematics for Secondary Schools A radian is a measure of an angle to 57.3°.
180° × 1
_______
= 57. 3° ,

90°. If s = 1, then θ =

π
Radian measure
which implies that, 1 radian is equivalent
formed at the centre of a circle by an
Angles can be used to measure the size
arc whose length is equal to that of the
of turning. Turns of the minute hand of a
radius of the same circle. The relationship
clock face and a wheel can be measured
between the arc length , central angle θ ,
in degrees or in radians. For instance, the
and radius can be used to compare the
minute hand of a clock face turns through
measurements of an angle in radians
π __
an angle of 90° or radians between
with measurements in degrees.
2
noon and 12:15 pm as shown in Figure
π rθ l πθ
Thus, if l = , then = . 5.5. The minute hand has covered an
180° r 180° angular distance of radians.
π __
2
118 Student\s Book Form Three
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MATHEMATIC F3 SB.indd 118
MATHEMATIC F3 SB.indd 118 18/09/2025 09:59:35