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S3
Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 11
Isosceles Triangles in Circles Right angle in a Semi-Circle
The CircleThe Circlew
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Tangent Line to a CircleDiameter Symmetry in a CircleCircumference of a CircleLength of an ARC of a CircleArea of a CircleArea of a SECTOR of a Circle
Summary of Circle Chapter
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 22
Starter QuestionsStarter Questionsw
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Q4. If a = 1 , b = 2 and c = 4
Find
Q3.
Q2. 30% of 200
( 25) ( 20)
5cm
2cm
3cm
4cm
2c - ab
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 33
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To identify isosceles triangleswithin a circle.
Aim of Today’s Lesson
Isosceles triangles Isosceles triangles in Circlesin Circles
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 44
When two radii are drawn to the ends of a chord, When two radii are drawn to the ends of a chord, An isosceles triangle is formed.An isosceles triangle is formed.
C
A B
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in Circlesin Circles
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 55
Special Properties of Isosceles TrianglesSpecial Properties of Isosceles Triangles
Two equal lengthsTwo equal lengths
Two equal anglesTwo equal angles
Angles in any triangle sum to 180Angles in any triangle sum to 180oo
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in Circlesin Circles
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 66
Q.Q.Find the angle xFind the angle xoo..
A
B
C
SolutionAngle at C is equal to:
Since the triangle is isosceleswe have
360 280 80o o o
xo
280o
2 80 180
180 80 2
2 100
50
o o o
o o o
o o
o o
x
x
x
x
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in Circlesin Circles
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 77
Begin Maths in Action Book page 181
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in Circlesin Circles
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 88
Starter QuestionsStarter Questionsw
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1. (-5) - (-6) + 11
2. Calculate 15% of 60
3. What is the perimeter and area
of a square with length 4cm.
4. Find the value of x
3x - 9 = 27
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 99
Semi-circle angleSemi-circle angle
To find the angle in a semi-circle made by a triangle with hypotenuse
equal to the diameter and the two smallerlengths meeting at the circumference.
Aim of Today’s Lesson
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1010
Semi-circle angleSemi-circle angle
Tool-kit requiredTool-kit required
1.1. ProtractorProtractor
2.2. PencilPencil
3.3. RulerRuler
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1111
1.1. Using your pencil trace roundUsing your pencil trace roundthe protractor so that you havethe protractor so that you have semi-circle.semi-circle.
2.2. Mark the Mark the centre of centre of the the semi-circle.semi-circle.
You should have You should have something like this.something like this.
Semi-circle angleSemi-circle anglew
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1212
Mark three points Mark three points
1.1. Outside the circleOutside the circle
x xx
x x
x x
x
x
Semi-circle angleSemi-circle angle
2. On the 2. On the circumferencecircumference3. Inside the circle3. Inside the circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1313
For each of the points For each of the points
Form a triangle by drawing aForm a triangle by drawing aline from each end of the line from each end of the diameter to the point.diameter to the point.Measure the angle at the Measure the angle at the various points.various points.
x
x
x
Semi-circle angleSemi-circle angle
Log your results in a table. Log your results in a table. InsideCircumferenc
eOutside
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1414
InsideCircumferenceOutside
x
Semi-circle angleSemi-circle angle
x
x
< 90o
> 90o
= 90= 90oo
Begin Maths in Action Book page 182
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1515
Starter QuestionsStarter Questionsw
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Write down as many equations as you can
e.g. a + b = 11
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1616
To understand what a tangent line isand its special property with the
radius at the point of contact.
Aim of Today’s Lesson
Tangent lineTangent linew
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1717
Tangent lineTangent line
A A tangent linetangent line is a line that is a line that touches a circle at touches a circle at only one point.only one point.
Which of theWhich of thelines are lines are tangent to tangent to the circle?the circle?
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1818
Tangent lineTangent line
The radius of the circle that touches the tangent The radius of the circle that touches the tangent line is called line is called the point of contact radius.the point of contact radius.
Special PropertySpecial Property
The point of contact radiusThe point of contact radiusis always perpendicular is always perpendicular
(right-angled)(right-angled)to the tangent line.to the tangent line.
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1919
Tangent lineTangent line
Q.Q.Find the length of the tangent line betweenFind the length of the tangent line between A and B.A and B.
A
B
8
10
C
SolutionRight-angled at A sinceAC is the radius at the pointof contact with the Tangent.
By Pythagoras Theorem we have2 2 2
2 2 2
2 2 2
2
8 10
10 8
100 64 36
36 6
a b c
a
a
a
a
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2020
Tangent lineTangent line
Begin Maths in Action Book page 185
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2121
Starter QuestionsStarter Questionsw
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Q4. If x = -1 , y = 3 and z = 10
Find
Q3.
Q2. 15% of 280
( 22) ( 29)
10cm
3cm
5cm
7cm
2z - 2xy
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2222
To understand some special properties
when a diameter bisects a chord.
Aim of Today’s Lesson
Diameter symmetryDiameter symmetryw
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2323
Diameter symmetryDiameter symmetry
A B
C
D
1. A line drawn through the centre of a circle through the midpoint a chord will ALWAYS cut
the chord at right-angles
2. A line drawn through the centre of a circle at right-angles to a chord willALWAYS bisect that chord.
3. A line bisecting a chord at right angles will ALWAYS pass through the centre of a circle.
O
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2424
Q.Q.Find the length of the chord A and B.Find the length of the chord A and B.
A
B
6O
Solution
By Pythagoras Theorem we have
2 2 2
2 2 2
2 2 2
2
6 10
10 6
100 36 64
64 8
a b c
a
a
a
a
Diameter symmetryDiameter symmetry
4
Since yellow line bisect AB and passesthrough centre O, triangle is right-angle.
Radius of the circle is 4 + 6 = 10.
10
Since AB is bisected The length of AB is
2 8 16ABlength
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2525
Begin Maths in Action Book page 187
Diameter symmetryDiameter symmetryw
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2626
Starter QuestionsStarter Questions
Q1.2 81Factorise x
Q2.12 4
Calculate 28 3
Q3. Calculate the area of the triangle
Q4.2Solve 2 3 0x x
12
8
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2727
Circumference Circumference of a circleof a circle
Aim of Today’s Lesson
To be able to use the formula for calculating
the circumference of a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2828
Main parts of the circle
radius
O
CircumferenceC D
Diameter2D r
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of a circleof a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2929
Q. Find the circumference of the circle ?
SolutionSolution
C D 4C
12.56C cm
4cm
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mCircumference Circumference
of a circleof a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3030
Q. The circumference of the circle is 60cm ? Find the length of the diameter and radius.
SolutionSolution
C D 60 D
60D cm
19 D cm
2D r
19 2r192
r
9.5r cm
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of a circleof a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3131
Now it’s your turn !
Begin Maths in Action Book page 190
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mCircumference Circumference
of a circleof a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3232
Starter QuestionsStarter Questions
Q3.
Q1. Calculate
72Simplif y
360
Q2. Make D the subject of the equation.
Q4. Calculate
5( 2) 2 ( 3) x x a
C D
2 2117 36
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3333
length of the length of the arc of a circlearc of a circle
Aim of Today’s Lesson
To find and be able to use the formula for calculating the length of an arc.
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3434
Q. What is an arc ?
A
B
AnswerAnswer
An arc is a fractionof the circumference.
minor arc
major arcww
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mlength of the length of the arc of a circlearc of a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3535
Q. Find the circumference of the circle ?
SolutionSolution
C D 20C
62.8C cm
10cm
Maths in Action Book Exercise 1 Questions 1 & 2 ww
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mlength of the length of the arc of a circlearc of a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3636
Q. Find the arc length for the semi-circle ?
Solution
C D length of circle circumf erencearc Fraction
31.4arc length cm
1
length ( )2
arc D180o
1
length ( 20)2
arc
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mlength of the length of the arc of a circlearc of a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3737
Q. Given part of the circle below. Find the arc length ?Solution
C D
1
( )4
arc length D
length 15.7arc cm
10cm length of circle circumf erencearc Fraction 90
o
1
( 20)4
arc length
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mlength of the length of the arc of a circlearc of a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3838
3 step process to calculating the length of an arc.
Step 1 Find the circumference of the circle.
Step 2 Find fraction of the whole circle.
Step 3 Find the arc length.
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mlength of the length of the arc of a circlearc of a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 3939
Solution
C D
1
( 12)8
arc length
length 4.71arc cm
6cm
length of circle circumf erencearc Fraction
45o
Q. Part of a circle is given below. Find the arc length.
45 1f raction=
360 8
o
o
Fraction of
circle
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mlength of the length of the arc of a circlearc of a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4040
3 step process to calculating the length of an arc.
Step 1 Find the circumference of the circle.
Step 2 Find fraction of the whole circle.
Step 3 Find the arc length.
C D
at centre360o
angle
length of circle circumf erencearc Fraction
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mlength of the length of the arc of a circlearc of a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4141
Q. Find the length of the arc AB below ?
9 cm
60o
A
B
SolutionSolution
1 : C = D
= 18
= 56.52cm
Step
60 1 2 : Fraction=
360 6Step
1 3 : arc length= 56.52 9.42
6Step cm
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4242
Now it’s your turn !
Begin Maths in Action Book page 193
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4343
Starter Questions Starter Questions
Q1. Find all missing angles
Q2. List the prime numbers between 10 and 20.
Q3. Solve
Q4. Calculate the area of the shape.
52o
3( 2) 5( 6)t t
10cm
5cm
4cm
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4444
The Area of a The Area of a circlecircle
Aim of Today’s Lesson
To come up with and be able to use the formula for calculating
the area of a circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4545
23
4
12
3
456
7
8
1 56
78
If we break the circleinto equal sectors
And lay them out side by side We get very close
to a rectangle.
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4646
12 3
45
67
8
thinner and thinnersectors
If we cut the sectors Thinner and thinner then we get closer and closer
to a rectangle. Hence we can represent the area of a circle
by a rectangle.
r
r
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4747
r
r
2Area of a circle r
Area of a rectangle l b 2Area of a rectangle r r r
But the area inside this rectangle is also the area of the circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4848
Q. Find the area of the circle ?
SolutionSolution
2A r 24A
250.26A cm
4cm
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4949
The Area The Area of a circleof a circle
Now it’s your turn !
Begin Maths in Action Book
Ex9.1 page 194 Q1-2
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5050
Q. The diameter of the circle is 60cm. Find area of the circle?
SolutionSolution
2A r60
302 2D
r cm
230A22827.43A cm
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5151
The Area The Area of a circleof a circle
Now it’s your turn !
Begin Maths in Action Book
Ex9.1 page 194 Q3-4
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5252
Q. The area of a circle is 12.64 cm2.
Find its radius?SolutionSolution
2A r
2 12.64
4r cm
4 2r cm
212.64 r
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5353
The Area The Area of a circleof a circle
Now it’s your turn !
Begin Maths in Action Book
Ex9.1 page 194 Q5 onwards
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5454
Starter QuestionsStarter Questions
Q3. Calculate
Q1. Factorise
2 51 25 8
Q2. Make h the subject of the equation.
Q4. The value of the standard deviation for petrolprices in Glasgow is 2.1 and 1.9 for Edinburgh.Comparing the two values write down a valid statement.
24 12 9 x x1 h
yt
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5555
Sector area of a circleSector area of a circle
Aim of Today’s Lesson
To find and be able to use the formula for calculating the sector of an circle.
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5656
Q. What is an sector area ?
Sector area of a circleSector area of a circle
A
B
AnswerAnswerSector area is a fraction
of the area of the whole circle.
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major sectorww
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5757
Sector area of a circleSector area of a circle
Q. Find the area of the circle ?
SolutionSolution
2A r 210A
2314A cm
10cm
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5858
Q. Find the sector area for the semi-circle ?
Solution
2A r of circle Area of circleSector Fraction
2157Sector cm
Sector area of a circleSector area of a circle
21( )
2Sector r180
o
21( 10 )
2Sector
10cm
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 5959
Q. Given part of the circle below. Find the sector area ?Solution
2A r
21( )
4Sector r
27.85Sector cm
Sector area of a circleSector area of a circle
c of circle Area of circleSe tor Fraction
21c ( 10 )
4Se tor
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6060
Sector area of a circleSector area of a circle
3 step process to calculating sector area of a circle.
Step 1 Find the area of the circle.
Step 2 Find fraction of the whole circle.
Step 3 Find the sector area.
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6161
Solution
2A r
1
( 36)8
Sector
2Sector 14.13cm
6cm
Sector area of a circleSector area of a circle
of circle area of circleSector Fraction
45o
Q. Part of a circle is given below. Find the sector area.
45 1f raction=
360 8
o
o
Fraction of
circle
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6262
Sector area of a circleSector area of a circle
3 step process for calculating the sector area.
Step 1 Find the area of the circle.
Step 2 Find fraction of the whole circle.
Step 3 Find the sector area.
2A r
at centre360o
angle
ec of circle area of circleS tor Fraction
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6363
Q. Find the area of the sector below ?
Sector area of a circleSector area of a circle
SolutionSolution
2
2
1 : A=
= 81
=254.34cm
Step r
60 1 2 : Fraction=
360 6Step
21 3 : arc length= 254.34 42.39
6Step cm
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6464
Sector area of a circleSector area of a circle
Now it’s your turn !
Begin Maths in Action Book page 196
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Starter QuestionsStarter Questions
Q1.Q1. Find all the missing anglesFind all the missing angles
Q2.Q2. CalculateCalculate
2 2 where a = 4 and b = (-5)a b
Q3. Q3. The cost of a new computer is £1000 +vat. If the vat is The cost of a new computer is £1000 +vat. If the vat is charged at 12% what is the total cost.charged at 12% what is the total cost.
Q4. Q4. The cost of a bag of sugar is £1.12. How much 50 bags The cost of a bag of sugar is £1.12. How much 50 bags cost.cost.
115o
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6666
Area is2A r
Summary of Circle Summary of Circle TopicTopic
Circumference is
C D
Sector area2 angle
Sector = 360
o
o
centrer
Arc length is
length
angleArc =
360
o
o
centreD
Diameter2D r
Radius12
r D
line that bisects a chord
1. Splits the chord into 2 equal halves.
2. Makes right-angle with the chord.
3. Passes through centre of the circle
Pythagoras TheoremSOHCAHTOA
Semi-circle angle is always 90
o
Tangent touches circle at
one pointand make angle 90
o with point of
contact radius
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Wednesday 19 April 2023Wednesday 19 April 2023 Created by Mr Lafferty Created by Mr Lafferty 6767
Summary of Circle TopicSummary of Circle Topic
Maths in Action Book Page 199Maths in Action Book Page 199
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