Www.mathsrevision.com National 4 EF 1.2a The Circle Circumference of the circle Diameter =...

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www.mathsrevision.c om National 4 EF 1.2a The Circle The Circle Circumference of the circle Diameter = Circumference ÷ π Area of a circle Composite Area Main Parts of a Circle Investigation of the Ratio of Circle Composite shapes Perimeter

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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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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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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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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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

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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

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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 38)

Composite Area Composite Area