Circles

Finding the radius of sectors Area of Segments Volumes of Cylinders Pi Circle words Rounding Refresher Circumference Area Perimeter and Area of compound shapes Perimeters of sectors Area of Sectors Finding the radius of sectors Area of Segments Volumes of Cylinders Volume of Spheres and cones Radius and Height of Cylinders Equation of a circle 1 Equation of a circle 2 Simultaneous Equations Circle theorems Circle formulae

Match the words to the definitions Sector Segment Chord Radius Arc Tangent Diameter Circumference The length around the outside of a circle A line which just touches a circle at one point A section of a circle which looks like a slice of pizza A section circle formed with an arc and a chord The distance from the centre of a circle to the edge The distance from one side of a circle to the other (through the centre) A section of the curved surface of a circle A straight line connecting two points on the edge of a circle HOME

Think about circles Think about a line around the outside of a circle Image that line straightened out- this is the circumference

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

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

How did you do? What do you think the world record is? 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 nevermore who in his wondrous lore passed on before left men his guidance how to circles mensurate.” Can you write one of your own? HOME

Rounding to Decimal Places 10 multiple choice questions

Round to 1 dp 0.34 0.4 0.3 A) B) 0.35 3 C) D)

Round to 1 dp 0.48 0.49 0.4 A) B) 0.47 0.5 C) D)

Round to 1 dp 2.75 2.8 2.74 A) B) 2.7 3.0 C) D)

Round to 1 dp 13.374 13.0 14.0 A) B) 13.3 13.4 C) D)

Round to 1 dp 26.519 25.0 26.6 A) B) 26.0 26.5 C) D)

Round to 2 dp 23.1782 23.20 23.18 A) B) 23.10 23.17 C) D)

Round to 2 dp 500.8251 500.80 500.8 A) B) 500.83 500.84 C) D)

Round to 3 dp 0.00417 0.00417 0.004 A) B) 0.00418 0.005 C) D)

Round to 2 dp 4.999 4.99 4.90 A) B) 5.00 4.98 C) D)

Round to 4 dp 0.72995 0.7300 0.7390 A) B) 0.7210 0.7399 C) D) HOME

Finding the Circumference You can find the circumference of a circle by using the formula- Circumference = π x diameter For Example- Area= π x 10 = 31.41592654.... = 31.4 cm (to 1 dp) 10cm

Circumference = π x diameter You can find the circumference of a circle by using the formula- Circumference = π x diameter For Example- Area= π x 10 = 31.41592654.... = 31.4 cm (to 1 dp) ANSWERS 10cm 1a 25.1cm b 12.6cm c 34.6cm d 66.0cm e 47.1cm Find the Circumference of a circles with: A diameter of : 8cm 4cm 11cm 21cm 15cm A radius of : 6cm 32cm 18cm 24cm 50cm 2a 37.7cm b 201.1cm c 113.1cm d 150.8cm e 157.1cm HOME

Finding the Area Area= π x Radius2 For Example- You can find the area of a circle by using the formula- Area= π x Radius2 For Example- Area= π x 72 = π x 49 = 153.93804 = 153.9 (to 1dp) cm2 7cm

Finding the Area Area= π x Radius2 2a 12.6 b 78.5 c 15.2 d 380.1 e You can find the area of a circle by using the formula- Area= π x Radius2 For Example- Area= π x 72 = π x 49 = 153.93804 = 153.9 (to 1dp) cm2 7cm ANSWERS 2a 12.6 b 78.5 c 15.2 d 380.1 e 314.2 f 153.9 g 100.5 h 28.3 HOME

Finding the Area of a Sector 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. For Example- The sector here is ¾ of a full circle Find the area of the full circle Area= π x 72 = π x 49 = 153.93804 = DON’T ROUND YET! Then find ¾ of that area ¾ of 153.93804 = 115.45353 (divide by 4 and multiply by 3) 7cm

Finding the Area of a Sector 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. For Example- The sector here is 3/5 of a full circle Find the area of the full circle Area= π x 72 = π x 49 = 153.93804 = DON’T ROUND YET! Then find 3/5 of that area 3/5 of 153.93804 = 92.362824 (divide by 5 and multiply by 3) = 92.4cm2 216° 7cm Sometimes the fraction cannot be simplified and will stay over 360

Finding the Area of a Sector The general formula for finding the area is: Area of sector= Angle of Sector x πr2 360 Fraction of full circle that sector covers “of” Area of full circle

Find the area of these sectors, to 1 decimal place Questions Find the area of these sectors, to 1 decimal place 3 ANSWERS 1 2 10cm 11cm 1 226.9 2 200.6 3 315.4 12cm 260° 190° 251° 4 19.0 5 61.2 6 80.7 6 4 5 32° 5cm 6.5cm 17cm 87° 166° HOME

Finding the Perimeter of a Sector 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 For Example- The sector here is ¾ of a full circle Find the area of the full circle Area= π x 14 (the diameter is twice the radius) = π x 49 = 43.982297...... = DON’T ROUND YET! Then find ¾ of that circumference ¾ of 43.982297...... = 32.99 cm (2 dp) Remember to add on 7 twice from the straight sides 7cm

Finding the Area of a Sector Sometimes 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 360 This sector is 250/360 or two hundred and fifty, three hundred and sixty-ITHS of the full circle Simplify if you can 250° Sometimes the fraction cannot be simplified and will stay over 360

Finding the Perimeter of a Sector The general formula for finding the area is: Perimeter of sector= (Angle of Sector x πd) + r + r 360 Fraction of full circle that sector covers “of” Circumference of full circle Don’t forget the straight sides This is the same as d of 2r, but I like r +r as it helps me remember why we do it

Find the perimeter of these sectors, to 1 decimal place Questions Find the perimeter of these sectors, to 1 decimal place ANSWERS 1 65.4 2 58.5 3 76.6 3 1 2 10cm 11cm 12cm 260° 190° 251° 4 17.6 5 31.8 6 43.5 6 4 5 32° 5cm 6.5cm 17cm 87° 166° HOME

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

Compound Area and Perimeter Find the perimeter of the shape below: 10cm Circumference of this semi circle = πd ÷ 2 = π x 10 ÷ 2 =15.7 cm (1dp) 8cm 10cm Perimeter of this rectangle= 8 + 8 + 10 =26cm (don’t include the red side) Perimeter of whole shape = 26 + 15.7 = 31.7 cm

Compound Area and Perimeter Find the areaof the shape below: 5cm Area of this quarter circle = π r2 ÷ 4 = π x 52 ÷ 4 = π x 25 ÷ 4 =19.7 cm2 (1dp) 10cm 11cm Area of whole shape = 110+ 19.7 = 129.7cm2 Area of this rectangle 10 x 11=110

Compound Area and Perimeter Work out all missing sides first Find the perimeter of the shape below: ? 5cm Circumference of this quarter circle = πd ÷ 4 = π x 10 ÷ 4 (if radius is 5, diameter is 10) =7.9 cm (1dp) 5cm 6cm 10cm 10cm 11cm Area of whole shape = 42+ 7.9 = 49.9cm Add all the straight sides= 10+10 + 11+ 5 + 6= 42cm

Find the perimeter and area of these shapes, to 1 decimal place Questions ANSWERS   AREA PERIMETER 1 38.1 23.4 2 135.0 61.3 3 181.1 60.8 Find the perimeter and area of these shapes, to 1 decimal place 2cm 3 1 2 4 27.3   5 129.3 47.7 6 128.5 12cm 10cm 20cm 6cm 11cm 4cm 17cm 4cm 6cm Do not worry about perimeter here Do not worry about perimeter here 6 4 5 10cm 10cm 5cm 5cm 5cm 12cm 20cm HOME

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 volume 1) Find the area of the circle 1) Multiple the area of the circle by the height or length of the cylinder

Volume of Cylinders 2 EXAMPLE- find the volume of this cylinder 4cm Find the area of the circle π x r2 π x 42 π x 16 = 50.3 cm2 (1dp) 10cm 2) Multiple the area of the circle by the height or length of the cylinder 50.3 (use unrounded answer from calculator) x 10 = 503cm3

Find the volume of these cylinders, to 1 decimal place Questions Find the volume of these cylinders, to 1 decimal place 3 ANSWERS 1 4cm 2 3cm 5cm 1 603.2 2 282.7 3 1178.1 12cm 10cm 15cm 4 142.0 5 2155.1 6 508.9 6 4 5 3cm 2cm 7cm 18cm 11.3cm 14cm HOME

Volume of Cylinders 2 EXAMPLE- find the height of this cylinder 4cm Find the area of the circle π x r2 π x 42 π x 16 = 50.3 cm2 (1dp) h 2) Multiple the area of the circle by the height or length of the cylinder 50.3 x h = 140cm3 Rearrange this to give h= 140 ÷ 50.3 h=2.8 cm Volume= 140cm3

Volume of Cylinders EXAMPLE- find the radius of this cylinder r Find the area of the circle π x r2 2) Multiple the area of the circle by the height or length of the cylinder π x r2 x 30 = 250cm3 94.2... x r2 = 250 Rearrange this to give r2 = 250 ÷ 94.2 r2 =2.7 (1dp) r= 1.6 (1dp) cm 30cm Volume= 250cm3

Find the volume of these cylinders, to 1 decimal place Questions ANSWERS 1 6.4 2 4.2 3 1.3 Find the volume of these cylinders, to 1 decimal place 4 2.3 5 1.8 6 1.9 3 1 4cm 2 3cm 5cm h h h volume= 320cm3 volume= 120cm3 volume= 100cm3 r 5 6 r 4 r 12cm 8cm 14cm volume= 90cm3 volume= 200cm3 volume= 150cm3 HOME

Volume of Spheres The formula for the volume of a sphere is e.g 10cm A= 4/3 x π x 103 A= 4/3 x π x 1000 A=4188.8 cm3 (1 dp) 10cm

Volume of Cones The formula for the volume of a cone is 10cm e.g A= 1/3 x π x 42 x 10 A= 1/3 x π x 16 x 10 A=167.6 cm3 (1 dp) 4cm

Find the volume of these spheres, to 1 decimal place Questions ANSWERS 1 4188.8 2 33510.3 3 523.6 Find the volume of these spheres, to 1 decimal place 4 201.1 5 122.5 6 1272.3 3 1 2 10cm 20cm 5cm 6 15cm 9cm 4 12cm 4cm 5 13cm 3cm HOME

Circles Theorems Angles connected by a chord Angle at the centre Triangles made with a diameter or radii Cyclic Quadrilaterals Tangents

Double Angle x Angle x = 50 x 2 x=100° The angle at the centre of a circle is twice the angle at the edge 50° x Example Angle x = 50 x 2 x=100°

2 3 1 25° 60° 90° Answers 1) 50 2)120 3)180 x x x 4)50 5)67.5 6)80 5 6 4 x x x 160° 100° 135° HOME

Triangles inside circles A triangle containing a diameter, will be a right angled triangle A triangle containing two radii, will be isosceles x 90°

2 3 1 Answers 1) X=30 2)x=18 3)x=45 4)X=40 y=40 5)x=30 y= 120 72° x x 60° x Answers 1) X=30 2)x=18 3)x=45 x x 4)X=40 y=40 5)x=30 y= 120 6)x=22 y=136 3 1 2 x y x 22° 100° y y 30° x HOME

Angles connected by a chord Angles connected by a chord are equal x x y y

2 3 1 Answers 1) x=25 y=15 2)x=125 y= 40 z=15 3)x=10 y=70 z=100 25° y 10° x x 40° z x 100° 125° y 15° z 15° Answers 1) x=25 y=15 2)x=125 y= 40 z=15 3)x=10 y=70 z=100 y 4)X=105 y=40 z=35 5)x=53 y= 30 z=72 6)x=85 y=80 z=17 5 4 z y x 80° 17° 95° 6 z y y 40° x x z 35° 25° 53° 30° HOME

Tangents to a circle A tangent will always meet a radius at 90° 90°

1 x 35 ° 1 140° x 2 120° x 4 40° x y z 3 z y HOME

Cyclic Quadrilaterals Opposite angles in a cyclic quadrilaterals add up to 180° 100 + y = 180 y=80° 60 + x = 180 x = 120 ° x y 100° 60°

2 1 3 Answers 1) x=70 y=85 2)x=126 y=105 3)x=100 y=160 4 5 95° 110° 54° 20° x y x 75° y 80° Answers 1) x=70 y=85 2)x=126 y=105 3)x=100 y=160 4 5 25° a 15° 70° w 4)w=15 x=70 y=65 z= 25 5)a=60 b=36 4b y z 2a x b HOME

Area of Segments Here we will look at finding the area of sectors You will need to be able to do two things: Find the area of a triangle using the formula- Area= ½ absinC Find the area of a sector using the formula- Area of sector= Angle of Sector x πr2 360 a C b

Example- find the area of the blue segment Step 1- find the area of the whole sector Area= 100/360 x π x r2 = 100/360 x π x 102 =100/360 x π x 100 =87.3cm2 Step 2- find the area of the triangle Area= ½ absinC =1/2 x 10 x 10 x sin100 = 49.2cm2 10cm 10cm 100° Step 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 Step 1- find the area of the whole sector Area= 120/360 x π x r2 = 120/360 x π x 122 =120/360 x π x 144 =150.8cm2 Step 2- find the area of the triangle Area= ½ absinC =1/2 x 12 x 12 x sin120 = 62.4cm2 12cm 12cm 120° Step 3- take the area of the triangle from the area of the segment 150.8 – 62.4 = 88.4 cm2

Find the area of the blue segments, to 1 decimal place Questions ANSWERS 1 75.1 2 29.5 3 201.1 Find the area of the blue segments, to 1 decimal place 4 8.3 5 51.8 6 33.0 3 1 2 10cm 85° 11cm 170° 130° 12cm 6 4 5 6.5cm 95° 5cm 65° 17cm 160° HOME

Finding the Radius or angle of a Sector 10cm 100° x Area=200cm2 Area=150 Area= 100 x π x r2 360 200= 100 x π x r2 200x360 = r2 100 x π 229.2=r2 15.1cm =r Area= θ x π x r2 360 150= θ x π x 102 150x360 = θ 102 x π 117.9°= θ

Find the missing radii and angles of these sectors, to 1 decimal place Questions ANSWERS 1 7.6 2 8.3 3 5.4 Find the missing radii and angles of these sectors, to 1 decimal place 4 160.4 5 122.1 6 47.6 3 1 2 r r r 200° 175° 250° Area=100cm2 Area=120cm2 Area=50cm2 6 4 5 5cm 6.5cm 17cm θ θ θ Area=120cm2 Area=35cm2 HOME Area=45cm2

The Equation of a Circle The general equation for a circle is (x-a)2 + (y-b)2=r2 This equation will give a circle whose centre is at (a,b) and has a radius of r For example a circle has the equation (x-2)2 + (y-3)2=52 This equation will give a circle whose centre is at (2,3) and has a radius of 5

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

The Equation of a Circle A circle has the equation (x-5)2 + (y-7)2=16 What is y when x is 1? (1-5)2 + (y-7)2=16 12+ (y-7)2=16 1+ (y-7)2=16 (y-7)2=15 y-7= ±3.9 (square root of 15 to 1 dp) y= 7±3.9 y= 10.9 or 3.1 There 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 Circle 1) 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=4 x2 + (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=16 4) Calculate the area and perimeter of the circle with the equation (x-3)2 + (y-5)2=16 5) 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=7 b) y when x=6 Answers 1a) 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) Answers 2) 16 3)Circumference = 25.1 Area=50.3 4)Circumference = 25.1 Area=50.3 5) Circles have the same radius but different centres, they are translations 6a) x= 11.5 or -7.5 b) y=11.3 or -3.3 HOME

The Equation of a Circle 2 Here we will look at rearranging equations to find properties of the circle they represent Remember- The general equation for a circle is (x-a)2 + (y-b)2=r2 The skill you will need is called completing the square, you may have used it to solve quadratic equations

The Equation of a Circle 2 Example x2 + y2 -6x – 8y =0 Create two brackets and put x in one and y in the other (x ) 2 + (y ) 2 = 0 Half 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= 0 Tidy this up (x -3) 2 + (y - 4) 2 – 25= 0 (x -3) 2 + (y - 4) 2 = 25 This circle has a radius of 5 and centre of (3,4)

The Equation of a Circle 2 Example x2 + y2 -10x – 4y- 7 =0 Create two brackets and put x in one and y in the other (x ) 2 + (y ) 2 = 0 Half 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= 0 Tidy this up (x -5) 2 + (y - 2) 2 – 36= 0 (x -5) 2 + (y - 2) 2 = 36 This circle has a radius of 6 and centre of (5,2)

The Equation of a Circle 2 You 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

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

Ways to solve quadratic equations- Simultaneous Equations A 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 x’s or y’s Because y=2x +1 we can rewrite the equation of the circle but instead of putting “y” in we’ll write “2x+1” So, (x-5)2 + (2x-1-7)2=16 (x-5)2 + (2x-8)2=16 expand the brackets x2-10x + 25 + 4x2 – 32x +64 = 16 simplify and make one side 0 5x2 -42x + 73=0 solve 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 question Ways to solve quadratic equations- Completing the square Factorising The Quadratic formula

Simultaneous Equations A 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 tangent 2 solutions to the quadratic 0 solutions to the quadratic the circle and the line never meet

Intercepts between lines and circles 1) Find out whether these circles and lines intercept, if they do find the coordinates of the interceptions (x-5)2 + (y-7)2=16 and y=3x-1 (x-3)2 + (y-8)2=36 and y=2x-2 (x+2)2 + (y-5)2=100 and y=3x + 3 (x+2)2 + (y+5)2=49 and 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) (2,2) (3.5,13.4) (-2.7,-5) (-0.4,3.3) (-6.4,-8.9) (-4.6,6.9) (-4.6,-8.9) HOME

Circle Formulae HOME