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Transcript of Www.mathsrevision.com National 4 EF 1.2a The Circle Circumference of the circle Diameter =...
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National 4EF 1.2a
The CircleThe Circle
Circumference of the circle
Diameter = Circumference
÷ πArea of a circle
Composite Area
Main Parts of a Circle
Investigation of the Ratio of Circle
Composite shapes Perimeter
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 22
Starter Questions Starter Questions
1. What shape has an infinte line symmetry.
2. Solve 3x+7 28.
3. Find the perimeter of the star.
4. Convert your answer to Q3 to (m).
7cm
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To revise the basics of the circle.
Main Parts of a Circle
1.1. Know the terms Know the terms circumference, diameter circumference, diameter and radius.and radius.
2.2. Identify them on a Identify them on a circle.circle.
3.3. Calculate the Calculate the circumference using circumference using formula.formula.
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 44
Main parts of the circle
Main part of a CircleMain part of a Circle
O
2D r
12
r D
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 55
Main part of a Main part of a CircleCircle
10cm
2cm
12
r D 5r cm
2D r 4D cm
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try Exercise 1Ch9 (page 100)
Main part of a CircleMain part of a Circle
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 77
Starter QuestionsStarter Questions
1. The diameter of a circle is 10cm. What is the radius.
2. The radius of a circle is 3.5cm. What is the diameter.
3. Find the perimeter of the star.
4. Convert your answer to Q3 to (mm).
5cm
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 88
Stars In Your EyesStars In Your Eyes
“Today Matthew we are going to be”
Archimedes of Syracuse
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 99
The Greek mathematician Archimedes of Syracuse (287- 212 BC) who flourished in Sicily is generally considered to be the greatest
mathematician of ancient times. He is credited with determining the relationship between the diameter and the circumference of a circle.
This was first recorded by Archimedes in the book Measurement of a Circle (225 BC).
In this investigation we are going to attempt to follow Archimedes stepsand arrive at the equation for determining the circumference for
any given circle.
History of CirclesHistory of Circles75 years old
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1010
Parts of the CircleParts of the Circle
Circumference
O
O = centre of circle
The radius is measured fromthe centre of the circle
to the edge.
radius
diameter
The diameter is measured from one edge to the other
passing through the centre of the circle.
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1111
Circle Circle InvestigationInvestigation
Construct a table shown below to enable usto record our results.
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1
2
3
4
5
6
7
8
Circle I nvestigation Calculations
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1212
Investigation of the CircleInvestigation of the Circle
To find a relationship between
the diameter and circumference of a given circle.
How can we measure the diameter and circumference
O
Our Investigation
Question ?
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1313
Measuring the Diameter Measuring the Diameter
of a circleof a circle
The diameter is the largest distance between one side of a circle to the other
passing through the centre O.
O
O Use a ruler
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1414
Measuring the Circumference Measuring the Circumference
of a circleof a circle
Flexible tape measure
Take a tape measure and put it round the circle
Read off the measurement
O
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1515
Measuring the Circumference of a Measuring the Circumference of a circlecircle
Roll along an even surface
One complete rotation equals the length of the circumference
Be careful to avoid slip!
Starting point
End point
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National 4EF 1.2a
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1 17.6 5.5
2 15.9 5.0
3 14.3 4.5
4 12.6 4.0
5 11.2 3.5
6 9.5 3.0
7 8.0 2.5
8
Circle I nvestigation Calculations
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1 17.6 5.5
2 15.9 5.0
3 14.3 4.5
4 12.6 4.0
5 11.2 3.5
6 9.5 3.0
7
8
Circle I nvestigation Calculations
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1 17.6 5.5
2 15.9 5.0
3 14.3 4.5
4 12.6 4.0
5 11.2 3.5
6
7
8
Circle I nvestigation Calculations
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1 17.6 5.5
2 15.9 5.0
3 14.3 4.5
4 12.6 4.0
5
6
7
8
Circle I nvestigation Calculations
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1 17.6 5.5
2 15.9 5.0
3 14.3 4.5
4
5
6
7
8
Circle I nvestigation Calculations
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1 17.6 5.5
2 15.9 5.0
3
4
5
6
7
8
Circle I nvestigation Calculations
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1 17.6 5.5
2
3
4
5
6
7
8
Circle I nvestigation Calculations
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1
2
3
4
5
6
7
8
Circle I nvestigation Calculations
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1 17.6 5.5
2 15.9 5.0
3 14.3 4.5
4 12.6 4.0
5 11.2 3.5
6 9.5 3.0
7 8.0 2.5
8 6.4 2.0
Circle I nvestigation Calculations
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1616
Circle InvestigationCircle Investigation
Circle Circumference = C Diameter = D C+D= C- D= C/D= CxD=
1 17.6 5.5 23.1 12.1 3.2 96.8
2 15.9 5.0 20.9 10.9 3.2 79.5
3 14.3 4.5 18.8 9.8 3.2 64.4
4 12.6 4.0 16.6 8.6 3.2 50.4
5 11.2 3.5 14.7 7.7 3.2 39.2
6 9.5 3.0 12.5 6.5 3.2 28.5
7 8.0 2.5 10.5 5.5 3.2 20.0
8 6.4 2.0 8.4 4.4 3.2 12.8
CalculationsCircle I nvestigation
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 1717
Circle InvestigationCircle Investigation
C D C circumference
D diameter
3.14
Using your results write down, in your own words, an approximate relationship between the circumference and diameter for a given circle.
“Circumference approximately equals three and bit diameters”Actual value is 3.14 which we write as π
Pronounced “Pi”
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
No Exercise
Circle InvestigationCircle Investigation
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questions
2. Calculate (-3m) x (-4m)
1. Name this shape.
3. Find the mean, median and mode of
data set 3,5,5,7,10
4
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To revise the basics of the circle.
Main Parts of a Circle
1.1. Know the terms Know the terms circumference, diameter circumference, diameter and radius.and radius.
2.2. Identify them on a Identify them on a circle.circle.
3.3. Calculate the Calculate the circumference using circumference using formula.formula.
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Main Parts of a Circle
Main parts of the circle
O
Circumference2D r
12
r D
C D
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Calculating the Calculating the CircumferenceCircumference
10cm
2cm
C D2D r 4D cm
Example : Find the length of the circumference (Perimeter)
of each circle
C D 10
31.4 cm 4
12.56 cm
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2323
Q. Calculate the curved part of this shape.Solution
C D
1curved length 12
4
length 9.4curved m
6m 1curved length
4D
90o
curved length 3
Calculating the Calculating the CircumferenceCircumference
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try Exercise 2Ch9 (page 100)
Main part of a CircleMain part of a Circle
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter Starter QuestionsQuestions
.
(-7) x (-12) 6
1. Find the circumference of the circle
2. Calculate
3. Find the mean, median and mode of
data set 6,7,3,10,8,10,12
4
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Recall knowledge of circles Recall knowledge of circles so far.so far.
1. To give some examples of problems that we can solve by applying our knowledge of circles and of the course so far.
Composite Composite PerimeterPerimeter
2.2. Solve mixed problems by Solve mixed problems by applying all our knowledge applying all our knowledge so far.so far.
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Things to think about when doing exercise.
The perimeter of a semicircle ……. Find whole circle then half it !
The perimeter of a quarter circle …….Find whole circle then quarter it !
Composite perimeter …….Find each perimeter and add them together !
Composite Composite PerimeterPerimeter
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2828
Q. Find the perimeter for the semi-circle shape ?
Solution
120
2P D
1
P 20 202
180o
P 10 20
P 51.4cm
Composite Composite PerimeterPerimeter
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 2929
Q. Find the perimeter of the shape below ?Solution
18 8
4P D
116 8 8
4P
28.6P cm
8cm
90o
4 16P
Composite Composite PerimeterPerimeter
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Example 1 : Find the perimeter of this shape
20cm
5 cm
Perimeter = 3 sides + semicircle
15 20 5 20
2 P
30 10 P
61 4.P cm
Composite Composite PerimeterPerimeter
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try Extension Booklet
6E (page 125)
Composite Composite PerimeterPerimeter
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter Starter QuestionsQuestions
2. Calculate (5d) x (-6d)
1. Name the shape.
3. Find the mean, median and mode of
data set 3,5,6,7,9
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1. To explain how we can find diameter of a circle if we know the circumference.
Finding the DiameterFinding the Diameter
The Circle
2.2. Solve diameter problems.Solve diameter problems.
1.1. Understand how to Understand how to rearrange circumference rearrange circumference formula to find diameter.formula to find diameter.
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
We can easily rearrange the circumference formula so that we have the diameter D on one side.
Finding the DiameterFinding the Diameter The Circle
You have 1 minute to rearrange equation.
C D
CD
Remember
change side change sign
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
C = 62.8 cm
C
D
Example : Find the diameter of each circle given the circumference.
62.8
20 cm5 cm
20cm
C =15.7 cm
C
D
15.7
5cm
Finding the DiameterFinding the Diameter The Circle
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try Extension Booklet
4E (page 32)
Finding the DiameterFinding the Diameter
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questions
2. Calculate (5b) x (-4b)
1. Name this shape.
3. 2x - 3 = 29
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National 4EF 1.2a
What do we call the distance AB in terms of the large circle.
Area of a CircleArea of a Circle
(2 )circumference r
( )radius r
O
A x
O
Ax B
What do we call the distance OA in terms ofthe large circle.
circumference
Peeling anorange
2 r
r
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
r
2 r
1
2A base height
2A r
What is the formula for the area of a right-angle triangle.
O
Ax
B
Use this formula to work outthe area for a circle.
12
2A r r
Area of a CircleArea of a Circle
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4040
23
4
1
2
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.
Area of a CircleArea of a Circle
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4141
12
34
56
78
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
Area of a CircleArea of a Circle
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4242
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
Area of a CircleArea of a Circle
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Area of a circleArea of a circle
Q. Find the area of the circle ?
SolutionSolution
2A r
24A250.26A cm
4cm
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National 4EF 1.2a
Tuesday 18 April 2023Tuesday 18 April 2023 Created by Mr Lafferty Created by Mr Lafferty 4444
The Area of a circleThe Area of a circle
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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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Area of a circleArea of a circle
Q. The diameter of the circle is 60cm. Find area of the circle?
SolutionSolution
2A r60
302 2
D
r cm
230A22826 A cm
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Now try Extension booklet
5E (page 35)
Area of a CircleArea of a Circle
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National 4EF 1.2a
20+8 4
1. Find the length of the curved part.
2. Calculate
3. Simplify 3(5y- 2)
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questions
2cm
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Recall knowledge of circles Recall knowledge of circles so far.so far.
1. To give some examples of problems that we can solve by applying our knowledge of circles and of the course so far.
Composite Area Composite Area
2.2. Solve mixed problems by Solve mixed problems by applying all our knowledge applying all our knowledge so far.so far.
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Example 1 : Find the area of the shape
20cm
5 cm
Area = rectangle + semicircle
21
2A l b r
2120 5 10
2( )A
100 157 A 2257 cmA
Composite Area Composite Area
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Example 1 : Find the area of the red part.
10cm
4cm
Area = Big Circle – Small Circle
2 2 - B SA r r2 25 - 2 A
21 A266 cmA
Composite Area Composite Area
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National 4EF 1.2a
18 Apr 202318 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Example 2 : A circle is contained in a square.Find the grey shaded area below.
Area = square - circle
2 2 -A l r28 8 4- ( )A
64 - 50.24A 213 76 cm.A
8cm
Composite Area Composite Area