SCHOLAR Study Guide
National 5 Mathematics
Course Materials
Topic 5: Arcs and sectors
Authored by:
Margaret Ferguson
Reviewed by:
Jillian Hornby
Previously authored by:
Eddie Mullan
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
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Distributed by the SCHOLAR Forum.
SCHOLAR Study Guide Course Materials Topic 5: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
Acknowledgements
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1
Topic 1
Arcs and sectors
Contents
5.1
Calculating the length of an arc . . . . . . . . . . . . . . . . . . . . . . . . . .
3
5.2
5.3
Finding the radius, diameter or angle given the length of an arc . . . . . . . .
Calculating the area of a sector . . . . . . . . . . . . . . . . . . . . . . . . . .
6
9
5.4
5.5
Finding the radius, diameter or angle given the area of a sector . . . . . . . .
Learning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
16
5.6
End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2
TOPIC 1. ARCS AND SECTORS
Learning objectives
By the end of this topic, you should be able to:
•
calculate the length of an arc;
•
calculate the radius, diameter or angle given the length of the arc;
•
calculate the area of a sector;
•
calculate the radius, diameter or angle given the area of the sector.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ARCS AND SECTORS
1.1
Calculating the length of an arc
Finding the circumference of a circle
Key point
Remember :
Circumference = π × Diameter ⇒ C = πD
Diameter = 2 × Radius ⇒ D = 2r
Therefore, the formula for circumference may also be written as:
Circumference = 2 × π × Radius ⇒ C = 2πr
Example
Problem:
A wheel has a diameter of 36 cm. What is its circumference?
Solution:
C = πD =
× 36 = 113 cm (to the nearest whole number)
..........................................
Q1:
A tart has a diameter of 12 cm. What is its circumference?
Round your answer to 1 decimal place.
..........................................
© H ERIOT-WATT U NIVERSITY
3
4
TOPIC 1. ARCS AND SECTORS
Finding the length of an arc of a circle
Key point
Remember:
Length of Arc =
angle
360◦
× π × D
Diameter = 2 r
By combining the calculations for a fraction of a quantity and the circumference of a
circle, we can find the length of an arc of a circle.
Example
Problem:
A fan has a radius of 30 cm. When fully
open it makes an angle of 160 ◦ at the
centre.
What is the length of the outer edge of
the fan?
Solution:
To find the length of the outer edge of the fan we must find a fraction of the
circumference of the whole circle.
The formula to find the length of an arc is angle /360◦ × πD
The circle has radius 30 cm so the diameter D = 2 × 30 = 60 cm.
The angle at the centre of the fan is 160 ◦ .
◦
The length of the outer edge of the fan = 160 /360◦ × π × 60 = 83 · 8 cm (to 1 d.p.).
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ARCS AND SECTORS
5
Example
Problem:
Look at the pie. What is the length of the
remaining crust?
Solution:
To find the length of the crust we must find a fraction of the circumference of the whole
circle.
The formula to find the length of an arc is angle /360◦ × πD
The circle has radius 3·5 cm so the diameter D = 3 · 5 × 2 = 7 cm.
The angle at the centre is 360 ◦ − 120◦ = 240◦ because the angle given is for the
piece of pie which has been eaten.
◦
The length of crust = 240 /360◦ × π × 7 = 14 · 7 cm (to 1 d.p.).
..........................................
Calculating the length of an arc practice
Q2:
A piece of cake has a radius of 12 cm.
It makes an angle of 40 ◦ at the centre.
What is the length of the outer edge of
the piece of cake? Give your answer to 1
decimal place.
..........................................
Q3:
A stuffed crust pizza has a radius of 21
cm.
The angle of the space where the slice
was removed is 128◦ .
What is the length of the remaining
stuffed crust? Give your answer to 1
decimal place.
..........................................
© H ERIOT-WATT U NIVERSITY
Go online
6
TOPIC 1. ARCS AND SECTORS
Calculating the length of an arc exercise
Q4:
Go online
A pie has diameter of 10 cm.
What is its circumference (in cm?)
..........................................
Q5:
A piece of cake has a radius of 10 cm.
The remaining cake has an angle of 190 ◦
at the centre.
What is the length, in cm, of the curved
edge of the piece of cake?
..........................................
Q6:
A stuffed crust pizza has a radius of 32
cm.
The angle of the space where the slice
was removed is 88◦ .
What is the length of the remaining
stuffed crust?
Give your answer to 1 decimal place.
..........................................
1.2
Finding the radius, diameter or angle given the length of
an arc
Key point
Remember:
Length of arc =
angle
360◦
× π × D
Diameter = 2 × Radius ⇒ D = 2r
By re-arranging the formula for the length of an arc we can find the angle or the diameter.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ARCS AND SECTORS
7
Examples
1.
Problem:
The length of the remaining icing around
the cake is 45 cm and the angle at the
centre is 260◦ .
Calculate the radius of the circular cake.
Solution:
The icing is an arc and is 45 cm long.
The angle at the centre of the remaining cake is 260 ◦ .
Remember the formula the length of an arc is length of arc =
If we replace what we know we get
260◦
260◦
×
π
×
D
×
π
=
2
·
269
45 =
360◦
360◦
angle /
360◦
45
45
2 · 269
D
=
2 · 269 × D
=
D
=
19 · 83252534
radius
=
D÷2
=
9 · 9 cm (to 1d.p.)
..........................................
× πD
(re − arrange the equation)
don t round yet
2.
Problem:
When a pendulum swings its path is an
arc.
When it swings from left to right the arc
has length 28 cm and the pendulum is 19
cm long.
Calculate the angle made as the
pendulum moves through its path.
Solution:
Length of arc = angle /360◦ × πD
The radius is 19 cm so the diameter = 2 × 19 = 38 and the length of the arc = 28.
If we replace what we know we get
© H ERIOT-WATT U NIVERSITY
8
TOPIC 1. ARCS AND SECTORS
28
=
28
=
angle
× π × 38
360◦
angle × 0 · 3316
=
angle
=
84 · 4◦ (to 1 d.p.)
..........................................
28
0 · 3316
angle
(π × 38 ÷ 360 = 0 · 3316)
(re − arrange the equation)
Finding the radius, diameter or angle practice
Q7:
Go online
If the length of the arc is 24 cm, calculate
the radius.
Give your answer to the nearest whole
number.
..........................................
Q8:
If the length of the arc is 251 m with a
radius of 200 m, calculate the angle x ◦ .
Give your answer to the nearest whole
number.
..........................................
Finding the radius, diameter or angle exercise
Q9:
Finding the diameter.
Go online
If the length of the arc is 10·28 cm,
calculate the diameter of the whole pizza.
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ARCS AND SECTORS
9
Q10: Finding the radius.
If the length of the arc is 109 m, calculate
the radius.
Give your answer to 1 decimal place.
..........................................
Q11: Finding the angle.
Calculate the angle at the x given that
the length of the arc is 221 m and the
radius is 39 m.
Give your answer to 1 decimal place.
..........................................
1.3
Calculating the area of a sector
Area of a circle
Key point
Remember :
Area = π r
2
Example
Problem:
A coin has a radius of 1·2 cm. What is the area of one side?
© H ERIOT-WATT U NIVERSITY
10
TOPIC 1. ARCS AND SECTORS
Solution:
A = πr 2 = π × 1 · 22 = 4 · 5 cm2 (to 1 decimal place)
..........................................
Q12:
A whole cheese has a radius of 16 cm. What is the area of one side?
Give your answer to the nearest whole number.
..........................................
Area of a sector of a circle
Key point
Remember:
Area of Sector =
angle
360◦
× π × r2
We can calculate the area of a sector of a circle by taking a fraction of the area of the
whole circle.
Example
Problem:
What area of the top of the pizza is
missing?
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ARCS AND SECTORS
11
Solution:
To find the area of the sector we must find a fraction of the area of the whole pizza
using the formula
area of sector = angle /360◦ × π × r 2
Since we know the angle and the radius we can put the values into the formula giving,
◦
area of sector = 120 /360◦ × π × 62 = 37 · 7 cm2 (to 1 d.p.)
..........................................
Calculating the area of a sector practice
Q13:
Go online
What area of the top of this 7 cm pizza is
missing?
Give your answer to 1 decimal place.
..........................................
Q14:
Calculate the area of the pizza remaining.
Give your answer to 1 decimal place.
..........................................
Calculating the area of a sector exercise
Q15: Area of a sector.
Go online
Calculate the area of the shaded sector
of the circle.
Give your answer to 1 decimal place.
..........................................
© H ERIOT-WATT U NIVERSITY
12
TOPIC 1. ARCS AND SECTORS
Q16: Area of a sector.
Calculate the area of the sector of pizza
which has been eaten.
Give your answer to 1 decimal place.
..........................................
Q17: Area of a sector.
What is the area of the pizza remaining
(in cm2 )?
Give your answer to 1 decimal place.
..........................................
1.4
Finding the radius, diameter or angle given the area of a
sector
Key point
Remember:
Area of Sector =
angle
360◦
× π × r2
By re-arranging the formula for the area of a sector we can find the angle or the radius.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ARCS AND SECTORS
13
Examples
1.
Problem:
The area of the top of the cake remaining
is 230 cm2 and the angle at the centre is
260◦ .
Calculate the radius of the circular cake
to 1 decimal place.
Solution:
The area of the sector is 230.
The angle at the centre of the remaining cake is 260 ◦ .
Remember the formula the area of a sector is
area of sector = angle /360◦ × π × r 2
If we replace what we know we get, 260◦
260◦
2
× π × r
× π = 2 · 269
230 =
360◦
360◦
230
230
2 · 269
r2
=
2 · 269 × r 2
=
r2
=
r
=
101 · 3662406
√
101 · 3662406
r
=
(re − arrange the equation)
(square root the answer)
10 · 1 cm (to 1 d.p.)
..........................................
2.
Problem:
This earring is in the shape of a slice of
cake.
The area of the top of the earring is 325
mm2 and the radius is 25 mm.
Calculate the angle at the point of the
earring to 1 decimal place.
Solution:
area of sector = angle /360◦ × π × r 2
If we replace the area of the sector and the radius we get,
© H ERIOT-WATT U NIVERSITY
14
TOPIC 1. ARCS AND SECTORS
325
=
325
325
5 · 454
x
=
angle
× π × 252
360◦
x × 5 · 454
=
x
=
59 · 6◦ (to 1 d.p.)
..........................................
π × 252 ÷ 360 = 5 · 454
(re − arrange the equation)
Finding the radius, diameter or angle given the area of a sector practice
Q18:
Go online
If the area of the cake is 4.3 inches2 ,
calculate the radius of this sector.
Calculate your answer to 1 decimal place.
..........................................
Q19:
If the area of the sector is 495 mm2 ,
calculate the angle y◦ .
Give your answer to the nearest degree.
..........................................
Finding the radius, diameter or angle given the area of a sector exercise
Q20: Finding the angle of a sector.
Go online
A wedge of cheese has an area of 440·5
cm2 and a radius of 29·5 cm. Calculate
the angle of the wedge of cheese.
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ARCS AND SECTORS
Q21: Finding the radius of a sector.
A sector has an angle of 318 ◦ . If the area
of the sector is 4 cm2 calculate the
radius.
Give your answer to 1 decimal place.
..........................................
Q22: Finding the diameter of a sector.
The area of the top of the cheese is 1309
cm2 . The angle of the remaining cheese
is 310◦ .
Calculate the diameter of the cheese.
..........................................
© H ERIOT-WATT U NIVERSITY
15
16
TOPIC 1. ARCS AND SECTORS
1.5
Learning Points
•
Circumf erence = πD and Diameter = 2rtherefore C = 2πr
•
Length of an Arc
•
Area of a Sector
•
To find an angle, radius or diameter when the length of an arc or the area of the
sector is known:
=
=
angle
360◦
angle
360◦
× π × Diameter
× π × radius2
◦
put the values you know into the formula;
◦
re-arrange or change the subject of the formula.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ARCS AND SECTORS
1.6
17
End of topic test
End of topic 5 test
Q23: Fraction of a circumference
Go online
A piece of cake has a radius of 100 cm. It
makes an angle of 120 ◦ at the centre.
What is the length, in cm, of the outer
edge of the piece of cake?
..........................................
Q24: Fraction of the area of a circle
A pizza has radius 8·1 cm and is missing
a sector with an angle of 230 ◦ .
What is the area of the pizza remaining
(in cm2 )?
..........................................
Q25: Finding the radius of an arc
If the length of the arc is 81 cm, calculate
the length of the radius.
Give your answer to the nearest whole
number.
..........................................
Q26: Finding an angle in a sector
If the area of the sector is 924 cm2 ,
calculate the size of the angle at the
centre of the pizza.
Give your answer to the nearest whole
number.
..........................................
© H ERIOT-WATT U NIVERSITY
18
TOPIC 1. ARCS AND SECTORS
Q27: Finding the diameter in a sector
The Pie Chart shows the occupational
structure in 1831.
The area of the sector labelled labourers
& servants is 40.2 cm2 with an angle
200◦ .
Calculate the diameter of the pie chart.
Give your answer to 1 decimal place.
..........................................
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 5
Answers to questions and activities
5 Arcs and sectors
Answers from page 3.
Q1: 37.7 cm
Calculating the length of an arc practice (page 5)
Q2: 8.4 cm
Q3: 85.0 cm
Calculating the length of an arc exercise (page 6)
Q4: 31·4 cm
Q5: 33 cm
Q6:
Diameter of the pizza = 2 × 32 = 64
Angle of the remaining pizza = 360-88 = 272
151·9 cm
Finding the radius, diameter or angle practice (page 8)
Q7: 13 cm
Q8: 72◦
Finding the radius, diameter or angle exercise (page 8)
Q9: 31 cm
Q10: 78·1 m
Q11: 324·7◦
Answers from page 10.
Q12: 804
Calculating the area of a sector practice (page 11)
Q13: 68.4 cm2
© H ERIOT-WATT U NIVERSITY
19
20
ANSWERS: TOPIC 5
Q14: 354.5 cm2
Calculating the area of a sector exercise (page 11)
Q15: 1165·7 cm2
Q16: 230·5 cm2
Q17: 241·3 cm2
Finding the radius, diameter or angle given the area of a sector practice (page 14)
Q18: 3.5 inches
Q19: 28◦
Finding the radius, diameter or angle given the area of a sector exercise (page
14)
Q20: 58◦
Q21: 1·2 cm
Q22: 44 cm
End of topic 5 test (page 17)
Q23: 209 cm
Q24: 74·4 cm2
Q25: 27 cm
Q26: 63◦
Q27: 9·6 cm
© H ERIOT-WATT U NIVERSITY