Circles

Circles

AreaCircumferenceArea of SectorsPerimeter and Area of compound shapesVolume of Spheres and conesRadius and Height of CylindersPerimeters of sectorsFinding the radius of sectorsPiCircle wordsVolumes of CylindersCircle theoremsRounding RefresherArea of SegmentsEquation of a circle 1Equation of a circle 2Simultaneous EquationsCircle formulae

Match the words to the definitions

SectorSegmentChordRadiusArcTangentDiameterCircumference

The length around the outside of a circleA line which just touches a circle at one pointA section of a circle which looks like a slice of pizzaA section circle formed with an arc and a chordThe distance from the centre of a circle to the edgeThe distance from one side of a circle to the other (through the centre)A section of the curved surface of a circleA straight line connecting two points on the edge of a circle

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Think about circles

Think about a line around the outside of a circleImage that line straightened out- this is the circumference

Pi

People noticed that if you divide the circumference of a circle by the diameter you ALWAYS get the same answerThey called the answer Pi () , which is:3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679 8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196 4428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273 724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609... You can use the button on your calculator

How many digits can you memorise in 2 minutes?3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609...

Write down pi!3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609... How did you do? What do you think the world record is?

Pi website

Pi Story One way to memorise Pi is to write a Pi-em (pi poem) where the number of letters in each word is the same as the number in pi. For example:

Now I, even I, would celebrate in rhymes inept,the great immortal Syracusan rivall'd nevermorewho in his wondrous lore passed on beforeleft men his guidance how to circles mensurate.

Can you write one of your own?

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Rounding to Decimal Places10 multiple choice questions

0.30.430.35A)B)C)D)Round to 1 dp0.34

0.50.490.40.47A)B)C)D)Round to 1 dp0.48

2.82.743.02.7A)B)C)D)Round to 1 dp2.75

13.413.014.013.3A)B)C)D)Round to 1 dp13.374

26.525.026.626.0A)B)C)D)Round to 1 dp26.519

23.1823.2023.1723.10A)B)C)D)Round to 2 dp23.1782

16

500.83500.80500.84500.8A)B)C)D)Round to 2 dp500.8251

0.0040.004170.0050.00418A)B)C)D)Round to 3 dp0.00417

18

5.004.994.984.90A)B)C)D)Round to 2 dp4.999

0.73000.73900.73990.7210A)B)C)D)Round to 4 dp0.72995HOME

Finding the CircumferenceYou can find the circumference of a circle by using the formula-

Circumference = x diameter

For Example-

Area= x 10 = 31.41592654.... = 31.4 cm (to 1 dp)

10cm

You can find the circumference of a circle by using the formula-

Circumference = x diameter

For Example-

Area= x 10 = 31.41592654.... = 31.4 cm (to 1 dp)

10cmFind the Circumference of a circles with:A diameter of :8cm4cm11cm21cm15cmA radius of :6cm32cm18cm24cm50cm

HOME1a25.1cmb12.6cmc34.6cmd66.0cme47.1cm

2a37.7cmb201.1cmc113.1cmd150.8cme157.1cm

ANSWERS

Finding the AreaYou can find the area of a circle by using the formula-

Area= x Radius2

For Example-

Area= x 72 = x 49 = 153.93804 = 153.9 (to 1dp) cm2

7cm

Finding the AreaYou can find the area of a circle by using the formula-

Area= x Radius2

For Example-

Area= x 72 = x 49 = 153.93804 = 153.9 (to 1dp) cm2

7cm

HOME2a12.6b78.5c15.2d380.1

e314.2f153.9g100.5h28.3

ANSWERS

Finding the Area of a SectorFor Example-The sector here is of a full circleFind the area of the full circle

Area= x 72 = x 49 = 153.93804 = DONT ROUND YET!

Then find of that area of 153.93804 = 115.45353 (divide by 4 and multiply by 3)

7cm

To find the area of a sector, you need to work out what fraction of a full circle you have, then work out the area of the full circle and find the fraction of that area.

Finding the Area of a SectorFor Example-The sector here is 3/5 of a full circleFind the area of the full circle

Area= x 72 = x 49 = 153.93804 = DONT ROUND YET!

Then find 3/5 of that area3/5 of 153.93804 = 92.362824 (divide by 5 and multiply by 3) = 92.4cm2

7cm

Sometimes it is not easy to see what fraction of a full circle you have.You can work it out based on the size of the angle. If a full circle is 360, and this sector is 216, the sector is 216/360, which can be simplified to 3/5.216Sometimes the fraction cannot be simplified and will stay over 360

Finding the Area of a SectorThe general formula for finding the area is:Area of sector= Angle of Sector x r2 360Fraction of full circle that sector coversofArea of full circle

Questions10cm

26011cm

19012cm

2515cm

876.5cm

16617cm

32Find the area of these sectors, to 1 decimal place123654HOME1226.92200.63315.4

419.0561.2680.7

ANSWERS

Finding the Perimeter of a SectorFor Example-The sector here is of a full circleFind the area of the full circle

Area= x 14 (the diameter is twice the radius) = x 49 = 43.982297...... = DONT ROUND YET!

Then find of that circumference of 43.982297...... = 32.99 cm (2 dp)

Remember to add on 7 twice from the straight sides

7cm

To find the perimeter of a sector, you need to work out what fraction of a full circle you have, then work out the circumference of the full circle and find the fraction of that circumference. You then need to add on the radius twice, as so far you have worked out the length of the curved edge

Finding the Area of a SectorSometimes you will not be able to see easily what fraction of the full circle you have.

To find the fraction you put the angle of the sector over 360Sometimes the fraction cannot be simplified and will stay over 360

250This sector is 250/360 or two hundred and fifty, three hundred and sixty-ITHS of the full circle

Simplify if you can

Finding the Perimeter of a SectorThe general formula for finding the area is:Perimeter of sector= (Angle of Sector x d) + r + r 360Fraction of full circle that sector coversofCircumference of full circleDont forget the straight sidesThis is the same as d of 2r, but I like r +r as it helps me remember why we do it

Questions10cm

26011cm

19012cm

2515cm

876.5cm

16617cm

32Find the perimeter of these sectors, to 1 decimal place123654HOMEANSWERS165.4258.5376.6

417.6531.8643.5

Here we will look at shapes made up of triangles, rectangles, semi and quarter circles.Find the area of the shape below:10cm8cm10cmArea of this rectangle= 8 x10=80cm2Area of this semi circle = r2 2= x 52 2= x 25 2=39.3 cm2 (1dp)Area of whole shape = 80 + 39.3 = 119.3 cm2Compound Area and Perimeter

Compound Area and Perimeter

Find the perimeter of the shape below:10cm8cm10cmPerimeter of this rectangle= 8 + 8 + 10=26cm(dont include the red side)Circumference of this semi circle = d 2= x 10 2=15.7 cm (1dp)Perimeter of whole shape = 26 + 15.7 = 31.7 cm

Compound Area and Perimeter

Find the areaof the shape below:11cm10cmArea of this quarter circle = r2 4= x 52 4= x 25 4=19.7 cm2 (1dp)Area of whole shape = 110+ 19.7 = 129.7cm25cmArea of this rectangle 10 x 11=110

Compound Area and Perimeter

Find the perimeter of the shape below:11cm10cmWork out all missing sides firstCircumference of this quarter circle = d 4= x 10 4 (if radius is 5, diameter is 10)=7.9 cm (1dp)Area of whole shape = 42+ 7.9 = 49.9cm5cm6cm5cm10cm?Add all the straight sides=10+10 + 11+ 5 + 6= 42cm

Questions10cm

11cm

12cm

6cm

20cm

10cmFind the perimeter and area of these shapes, to 1 decimal place123654HOME

4cm17cm20cm2cm6cm4cm5cm12cm10cm5cm5cmDo not worry about perimeter hereDo not worry about perimeter hereANSWERSAREAPERIMETER138.123.42135.061.33181.160.8

427.35129.347.76128.5

Volume of Cylinders

Here we will find the volume of cylinders

Cylinders are prisms with a circular cross sections, there are two steps to find the volume1) Find the area of the circle1) Multiple the area of the circle by the height or length of the cylinder

Volume of Cylinders 2

Find the area of the circle x r2 x 42 x 16 = 50.3 cm2 (1dp) 2) Multiple the area of the circle by the height or length of the cylinder

50.3 (use unrounded answer from calculator) x 10 = 503cm3EXAMPLE- find the volume of this cylinder10cm4cm

QuestionsFind the volume of these cylinders, to 1 decimal place123654

4cm12cm

3cm10cm

5cm15cm

3cm18cm

7cm14cm

2cm11.3cmHOME1603.22282.731178.1

ANSWERS4142.052155.16508.9

Volume of Cylinders 2

Find the area of the circle x r2 x 42 x 16 = 50.3 cm2 (1dp) 2) Multiple the area of the circle by the height or length of the cylinder

50.3 x h = 140cm3 Rearrange this to giveh= 140 50.3h=2.8 cm EXAMPLE- find the height of this cylinderVolume= 140cm34cmh

Volume of Cylinders

Find the area of the circle x r22) Multiple the area of the circle by the height or length of the cylinder

x r2 x 30 = 250cm394.2... x r2 = 250Rearrange this to giver2 = 250 94.2r2 =2.7 (1dp)r= 1.6 (1dp) cm EXAMPLE- find the radius of this cylinderVolume= 250cm3r30cm

QuestionsFind the volume of these cylinders, to 1 decimal place123654

4cmh

3cmh

5cmh

r8cm

r14cm

r12cmvolume= 100cm3volume= 120cm3volume= 320cm3volume= 200cm3volume= 150cm3volume= 90cm3HOMEANSWERS16.424.231.3

42.351.861.9

Volume of SpheresThe formula for the volume of a sphere is 10cm

e.g

A= 4/3 x x 103A= 4/3 x x 1000A=4188.8 cm3 (1 dp)

Volume of ConesThe formula for the volume of a cone is 10cm

e.g

A= 1/3 x x 42 x 10A= 1/3 x x 16 x 10A=167.6 cm3 (1 dp)

4cm

10cmQuestionsFind the volume of these spheres, to 1 decimal place123654HOME20cm5cm

12cm4cm

13cm3cm

15cm9cm14188.8233510.33523.6

ANSWERS4201.15122.561272.3

Circles TheoremsAngle at the centreAngles connected by a chordTriangles made with a diameter or radiiCyclic QuadrilateralsTangents

50xExampleDouble AngleThe angle at the centre of a circle is twice the angle at the edgeAngle x = 50 x 2 x=100

25x1601006013590xxxxx123645HOMEAnswers1) 502)1203)180

4)505)67.56)80

90Triangles inside circlesA triangle containing a diameter, will be a right angled triangle

A triangle containing two radii, will be isosceles

xx

60x123312

72xxx

xyyx100x3022yAnswers1) X=302)x=183)x=45

4)X=40 y=405)x=30 y= 1206)x=22 y=136xHOME

Angles connected by a chordAngles connected by a chord are equal

xxyy

25x123645

y15yzzxyxzxyyzx255330

zyx8017953540125154010100Answers1) x=25 y=152)x=125 y= 40 z=153)x=10 y=70 z=100

4)X=105 y=40 z=355)x=53 y= 30 z=726)x=85 y=80 z=17HOME

90Tangents to a circleA tangent will always meet a radius at 90

40xyz3

120x41

140x2

x35 1yzHOME

Cyclic QuadrilateralsOpposite angles in a cyclic quadrilaterals add up to 180

xy10060100 + y = 180 y=80

60 + x = 180 x = 120

xyxy

xy9511054752080

x2a4b1570ab125yzw2345Answers1) x=70 y=852)x=126 y=1053)x=100 y=160

4)w=15 x=70 y=65 z= 255)a=60 b=36HOME

Here we will look at finding the area of sectorsYou will need to be able to do two things:Area of SegmentsFind the area of a sector using the formula-

Find the area of a triangle using the formula-

Area= absinC

Area of sector= Angle of Sector x r2 360

Cba

Example-find the area of the blue segment

10cm10cm

100Step 1- find the area of the whole sectorArea= 100/360 x x r2 = 100/360 x x 102 =100/360 x x 100 =87.3cm2Step 2- find the area of the triangleArea= absinC =1/2 x 10 x 10 x sin100 = 49.2cm2Step 3- take the area of the triangle from the area of the segment

87.3 49.3 = 38 cm2

Example-find the area of the blue segment

12cm12cm

120Step 1- find the area of the whole sectorArea= 120/360 x x r2 = 120/360 x x 122 =120/360 x x 144 =150.8cm2Step 2- find the area of the triangleArea= absinC =1/2 x 12 x 12 x sin120 = 62.4cm2Step 3- take the area of the triangle from the area of the segment

150.8 62.4 = 88.4 cm2

Questions10cm13011cm8512cm1705cm956.5cm

Find the area of the blue segments, to 1 decimal place123

654HOME17cm65160ANSWERS175.1229.53201.1

48.3551.8633.0

Finding the Radius or angle of a Sector

r

Area= 100 x x r2 360200= 100 x x r2 360

200x360 = r2100 x

229.2=r215.1cm =r

Area=15010cmx100Area=200cm2Area= x x r2 360150= x x 102 360

150x360 = 102 x

117.9=

Questionsr

200r

175r

2505cm

6.5cm

17cm

Find the missing radii and angles of these sectors, to 1 decimal place123654HOMEArea=100cm2Area=120cm2Area=50cm2Area=35cm2Area=45cm2Area=120cm2ANSWERS17.628.335.4

4160.45122.1647.6

The Equation of a CircleThe general equation for a circle is (x-a)2 + (y-b)2=r2This equation will give a circle whose centre is at (a,b) and has a radius of rFor example a circle has the equation (x-2)2 + (y-3)2=52This equation will give a circle whose centre is at (2,3) and has a radius of 5

The Equation of a CircleA circle has the equation (x-5)2 + (y-7)2=16This equation will give a circle whose centre is at (5,7) and has a radius of 4 (square root of 16 is 4)For example a circle has the equation (x+2)2 + (y-4)2=100This equation will give a circle whose centre is at (-2,4) and has a radius of 10You could think of this as (x - -2)2

The Equation of a CircleA circle has the equation (x-5)2 + (y-7)2=16What is y when x is 1?

(1-5)2 + (y-7)2=1612+ (y-7)2=161+ (y-7)2=16(y-7)2=15y-7= 3.9 (square root of 15 to 1 dp)y= 73.9y= 10.9 or 3.1There are two coordinates on the circle with x=1, one is (1,10.9) and the other is (1,3.1)

The Equation of a Circle1) Write down the coordinates of the centre point and radius of each of these circles:(x-5)2 + (y-7)2=16(x-3)2 + (y-8)2=36(x+2)2 + (y-5)2=100(x+2)2 + (y+5)2=49(x-6)2 + (y+4)2=144 x2 + y2=4x2 + (y+4)2=121 (x-1)2 + (y+14)2 -16=0 (x-5)2 + (y-9)2 -10=15 2) What is the diameter of a circle with the equation (x-1)2 + (y+3)2 =64 3) Calculate the area and circumference of the circle with the equation (x-5)2 + (y-7)2=164) Calculate the area and perimeter of the circle with the equation (x-3)2 + (y-5)2=165) Compare your answers to question 3 and 4, what do you notice, can you explain this?6 ) A circle has the equation (x+2)2 + (y-4)2=100, find:a) x when y=7b) y when x=6HOMEAnswers1a) r=4 centre (5,7)b) r=6 centre (3,8)c) r=4 centre (-2,5)d) r=10 centre (-2,-5)e) r=7 centre (6,-4)f) r=12 centre (0,0)g) r=411centre (0,-4)h) r=4 centre (1,-14)i) r=5 centre (5,9)6a) x= 11.5 or -7.5b) y=11.3 or -3.3Answers2) 163)Circumference = 25.1 Area=50.34)Circumference = 25.1 Area=50.35) Circles have the same radius but different centres, they are translations

The Equation of a Circle 2Remember- The general equation for a circle is (x-a)2 + (y-b)2=r2The skill you will need is called completing the square, you may have used it to solve quadratic equationsHere we will look at rearranging equations to find properties of the circle they represent

The Equation of a Circle 2Example x2 + y2 -6x 8y =0 Create two brackets and put x in one and y in the other(x ) 2 + (y ) 2 = 0Half the coefficients of x and y and put them into the brackets, and then subtract those numbers squared(x -3) 2 + (y - 4) 2 32 - 42= 0Tidy this up(x -3) 2 + (y - 4) 2 25= 0(x -3) 2 + (y - 4) 2 = 25This circle has a radius of 5 and centre of (3,4)

The Equation of a Circle 2Example x2 + y2 -10x 4y- 7 =0 Create two brackets and put x in one and y in the other(x ) 2 + (y ) 2 = 0Half the coefficients of x and y and put them into the brackets, and then subtract those numbers squared(x -5) 2 + (y - 2) 2 52 22 - 7= 0Tidy this up(x -5) 2 + (y - 2) 2 36= 0(x -5) 2 + (y - 2) 2 = 36This circle has a radius of 6 and centre of (5,2)

The Equation of a Circle 2You must always make sure the coefficient of x2 and y2 is 1

You may have to divide through 2x2 + 2y2 -20x 8y- 14 =0

Divide by 2 to give x2 + y2 -10x 4y- 7 =0

Then put into the form x2 + y2 -10x 4y- 7 =0

QuestionsPut this equations into the form (x-a)2 + (y-b)2=r2 then find the centre and radius of the circle x2 + y2 -8x 4y- 5 =0x2 + y2 -12x 6y- 4 =0 x2 + y2 -4x 10y- 20 =0 x2 + y2 -10x 14y- 7 =0 x2 + y2 -12x 2y- 62 =02x2 +2y2 -20x 20y- 28 =0 3x2 + 3y2 -42x 24y- 36 =0 5x2 + 5y2 -100x 30y- 60 =0 HOMEAnswers1) r=5 centre (4,2)2) r=7 centre (6,3)3) r=7 centre (2,5)4) r=9 centre (5,7)Answers5) r=10 centre (6,1)6) r=8 centre (5,5)7) r=8 centre (6,4)8) r=11 centre (10,3)

Simultaneous EquationsA circle has the equation (x-5)2 + (y-7)2=16 and a line has an equation of y=2x+1, at what points does the line intercept the circle?We need to substitute into the equation of the circle so that we only have xs or ys

Because y=2x +1 we can rewrite the equation of the circle but instead of putting y in well write 2x+1

So, (x-5)2 + (2x-1-7)2=16(x-5)2 + (2x-8)2=16expand the bracketsx2-10x + 25 + 4x2 32x +64 = 16 simplify and make one side 05x2 -42x + 73=0solve this quadratic equation to find x, Put the value / values of x into y=2x+1 to find the coordinates of the intercept / intercepts to answer the questionWays to solve quadratic equations-Completing the squareFactorisingThe Quadratic formula

73

Simultaneous EquationsA quadratic equation can give 1,2 or no solutions, a line can cross a circle at 1,2 or no points

1 solution to the quadratic-The line is a tangent0 solutions to the quadratic the circle and the line never meet2 solutions to the quadratic

74

Intercepts between lines and circles1) Find out whether these circles and lines intercept, if they do find the coordinates of the interceptions(x-5)2 + (y-7)2=16and y=3x-1(x-3)2 + (y-8)2=36and y=2x-2(x+2)2 + (y-5)2=100and y=3x + 3(x+2)2 + (y+5)2=49and 2y+4=x(x-6)2 + (y+4)2=144 and y -3x =5

ANSWERS (all have been rounded)(3.6,9.8)and(2.2,5.6)(7.2,12.3)and(2,2)(3.5,13.4)and(-2.7,-5)(-0.4,3.3)and(-6.4,-8.9)(-4.6,6.9)and(-4.6,-8.9)

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Circle FormulaeHOME