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1
Topic 5
Arcs and Sectors
Contents
5
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5
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2
4
5
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5
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7
9
5
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12
5
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13
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
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ARCS AND SECTORS
5
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What is its circumference?
Solution:
C = πD = π × 36 = 113 cm (to the nearest whole number)
...
What is its circumference?
Round your answer to 1 decimal place
...
Finding the Length of an Arc of a Circle
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
...
Problem:
A fan has a radius of 30 cm
...
What is the length of the outer edge of
the fan?
Solution:
The diameter = 2 × 30 = 60 cm
...
p
...
© H ERIOT-WATT U NIVERSITY
TOPIC 5
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Problem:
Look at the pie
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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
...
◦
The length of crust = 240 /360◦ × π × 7= 14·7 cm (to 1 d
...
...
It makes an angle of 40 ◦ at the centre
...
...
The angle of the space where the slice
was removed is 128◦
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ARCS AND SECTORS
Calculating the length of an arc exercise
Q4:
A pie has diameter of 10 cm
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Q5:
A piece of cake has a radius of 10 cm
...
What is the length, in cm, of the outer
edge of the piece of cake?
...
2
Finding the radius, diameter or angle given the length of
an arc
By re-arranging the formula for the length of an arc we can find the angle or the diameter
...
Problem:
The length of the remaining icing is 45
cm and the angle at the centre is 260 ◦
...
Solution:
The icing is an arc and is 45 cm long
...
Remember the formula the length of an arc is length of arc =
If we replace what we know we get
angle /
360◦
× πD
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TOPIC 5
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p
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2
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When it swings from left to right the arc
has length 28 cm and the pendulum is 19
cm long
...
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
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p
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Finding the radius, diameter or angle practice
Q6:
If the length of the arc is 24 cm, calculate
the radius
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ARCS AND SECTORS
Q7:
If the length of the arc is 251 m with a
radius of 200 m, calculate the angle x ◦
...
...
If the length of the arc is 10·28 cm,
calculate the diameter of the whole pizza
...
Q9:
Finding the radius
...
Give your answer to 1 decimal place
...
Q10: Finding the angle
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Give your answer to 1 decimal place
...
© H ERIOT-WATT U NIVERSITY
TOPIC 5
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3
Calculating the area of a sector
Area of a Circle
Key Point
Remember: Area = π r
2
Example
Problem:
A disc has a radius of 12 cm
...
Q11:
A cheese has a radius of 16 cm
...
...
Example
Problem:
What area of the top of the pizza is
missing?
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 × r2
Since we know the angle and the radius we can put the values into the formula giving,
◦
area of sector = 120 /360◦ × π × 62 = 7 cm2 (to 1 d
...
)
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ARCS AND SECTORS
Calculating the area of a sector practice
Q12:
What area of the top of this 7 cm pizza is
missing?
Give your answer to 1 decimal place
...
Q13:
Calculate the area of the Pizza
remaining
...
...
Calculate the area of the shaded sector
of the circle
...
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ARCS AND SECTORS
Q15: Area of a sector
...
...
Calculate the area of the sector of pizza
which has been eaten
...
...
5
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Examples
1
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Calculate the radius of the circular cake
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The angle at the centre of the remaining cake is 260 ◦
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ARCS AND SECTORS
If we replace what we know we get
230 =
230◦
× π × r2
360◦
230◦
× π = 2 · 269
360◦
(re − arrange the equation)
230 = 2 · 269 × r 2
230
= r2
2 · 269
r 2 = 101 · 3662406 (square root the answer)
√
r = 101 · 3662406
r = 10 · 1 cm
...
Problem:
This earring is in the shape of a slice of
cake
...
Calculate the angle at the point of the
earring
...
p
...
Finding the radius, diameter or angle given the area of a sector practice
Q17:
If the area of the cake is 4
...
...
ARCS AND SECTORS
Q18:
If the area of the sector is 495 mm2 ,
calculate the angle y◦
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Finding the radius, diameter or angle given the area of a sector exercise
Q19: Finding the angle of a sector
...
...
Calculate the angle of the sector marked
p◦
...
...
The area of the top of the cheese is 1309
cm2
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ARCS AND SECTORS
5
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TOPIC 5
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6
End of topic test
End of topic 5 test
Q22: Fraction of a Circumference
A piece of cake has a radius of 100 cm
...
What is the length, in cm, of the outer
edge of the piece of cake?
Give your answer to the nearest whole
number
...
Q23: Fraction of the Area of a Circle
What is the area of the pizza remaining
(in cm2 )?
Give your answer to 1 decimal place
...
Q24: 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
...
© H ERIOT-WATT U NIVERSITY
13
14
TOPIC 5
...
The area of the sector labelled labourers
& servants is 40
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Calculate the diameter of the pie chart
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ANSWERS: TOPIC 5
Answers to questions and activities
5 Arcs and Sectors
Answers from page 2
...
Q11: 804 cm2
Calculating the area of a sector practice (page 8)
Q12: 68·4 cm2
Q13: 354·5 cm2
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15
16
ANSWERS: TOPIC 5
Calculating the area of a sector exercise (page 8)
Q14: 1165·7 cm 2
Q15: 241·3 cm2
Q16: 230·5 cm2
Finding the radius, diameter or angle given the area of a sector practice (page 10)
Q17: 3·5 inches2
Q18: 28◦
Finding the radius, diameter or angle given the area of a sector exercise (page
11)
Q19: 58◦
Q20: 318·3◦
Q21: 44 cm
End of topic 5 test (page 13)
Q22: 209 cm
Q23: 74·4 cm2
Q24: 27 cm
Q25: 63◦
Q26: 9·6 cm
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