circles. area circumference area of sectors perimeter and area of compound shapes perimeter and area...
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Circles
Area
Circumference
Area of SectorsPerimeter and
Area of compound shapes
Volume of Spheres and
cones
Radius and Height of Cylinders
Perimeters of sectors
Finding the radius of sectors
Pi Circle words
Volumes of Cylinders
Circle theorems
Rounding Refresher
Area of Segments
Equation of a circle 1
Equation of a circle 2
Simultaneous Equations
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
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Think about circlesThink 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 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...
Write down pi!
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609...
How did you do? What do you think the world record is?
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 Places
10 multiple choice questions
0.30.4
30.35
A) B)
C) D)
Round to 1 dp
0.34
0.5
0.49 0.4
0.47
A) B)
C) D)
Round to 1 dp
0.48
2.8 2.74
3.02.7
A) B)
C) D)
Round to 1 dp
2.75
13.4
13.0 14.0
13.3
A) B)
C) D)
Round to 1 dp
13.374
26.5
25.0 26.6
26.0
A) B)
C) D)
Round to 1 dp
26.519
23.1823.20
23.1723.10
A) B)
C) D)
Round to 2 dp
23.1782
500.83
500.80
500.84
500.8A) B)
C) D)
Round to 2 dp
500.8251
0.0040.00417
0.0050.00418
A) B)
C) D)
Round to 3 dp
0.00417
5.00
4.99
4.98
4.90A) B)
C) D)
Round to 2 dp
4.999
0.7300 0.7390
0.73990.7210
A) B)
C) D)
Round to 4 dp
0.72995
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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
You can find the circumference of a circle by using the formula-
Circumference = π x diameterFor Example-
Area= π x 10 = 31.41592654.... = 31.4 cm (to 1 dp)
10cm
Find the Circumference of a circles with:1. A diameter of :
a) 8cmb) 4cmc) 11cmd) 21cme) 15cm
2. A radius of :a) 6cmb) 32cmc) 18cmd) 24cme) 50cm HOME
1a 25.1cmb 12.6cmc 34.6cmd 66.0cme 47.1cm
2a 37.7cmb 201.1cmc 113.1cmd 150.8cme 157.1cm
ANSWERS
Finding the Area
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 AreaYou 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
HOME
2a 12.6b 78.5c 15.2d 380.1e 314.2f 153.9g 100.5h 28.3
ANSWERS
Finding the Area of a Sector
For Example-The sector here is ¾ of a full circleFind the area of the full circle
Area= π x 72
= π x 49 = 153.93804 = DON’T ROUND YET!
Then find ¾ of that area¾ 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 Sector
For Example-The sector here is 3/5 of a full circleFind 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 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.
216°
Sometimes 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 πr2
360
Fraction of full circle that sector
covers“of” Area of full
circle
Questions
10cm
260°
11cm
190°
12cm
251°
5cm87°
6.5cm
166°
17cm32°
Find the area of these sectors, to 1 decimal place
1 2 3
654
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1 226.92 200.63 315.44 19.05 61.26 80.7
ANSWERS
Finding the Perimeter of a Sector
For Example-The sector here is ¾ of a full circleFind 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 circumference¾ 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 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
Sometimes the fraction cannot be simplified and will stay over 360
250°
This 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 πd) + r + r
360
Fraction of full circle that sector covers “of” Circumference of full
circleDon’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
Questions
10cm
260°
11cm
190°
12cm
251°
5cm87°
6.5cm
166°
17cm32°
Find the perimeter of these sectors, to 1 decimal place
1 2 3
654
HOME
ANSWERS1 65.42 58.53 76.64 17.65 31.86 43.5
Here we will look at shapes made up of triangles, rectangles, semi and quarter circles.
Find the area of the shape below:
10cm
8cm
10cm
Area of this rectangle= 8 x10
=80cm2
Area of this semi circle = π r2 ÷ 2= π x 52 ÷ 2= π x 25 ÷ 2=39.3 cm2 (1dp)
Area of whole shape = 80 + 39.3 = 119.3 cm2
Compound Area and Perimeter
Compound Area and Perimeter
Find the perimeter of the shape below:
10cm
8cm
10cm
Perimeter of this rectangle= 8 + 8 + 10
=26cm(don’t include the red side)
Circumference of this semi circle = πd ÷ 2= π 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:
11cm
10cm
Area of this quarter circle = π r2 ÷ 4= π x 52 ÷ 4= π x 25 ÷ 4=19.7 cm2 (1dp)
Area of whole shape = 110+ 19.7 = 129.7cm2
5cm
Area of this rectangle 10 x 11=110
Compound Area and Perimeter
Find the perimeter of the shape below:
11cm
10cm
Work out all missing sides first
Circumference of this quarter circle = πd ÷ 4= π x 10 ÷ 4 (if radius is 5, diameter is 10)=7.9 cm (1dp)
Area of whole shape = 42+ 7.9 = 49.9cm
5cm
6cm5cm
10cm
?
Add all the straight sides=10+10 + 11+ 5 + 6= 42cm
Questions
10cm
11cm
12cm
6cm
20cm
10cm
Find the perimeter and area of these shapes, to 1 decimal place
1 2 3
654
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4cm
17cm
20cm
2cm
6cm
4cm
5cm
12cm
10cm
5cm5cm
Do not worry about perimeter here
Do not worry about perimeter here
ANSWERS AREA PERIMETER
1 38.1 23.42 135.0 61.33 181.1 60.84 27.3
5 129.3 47.76 128.5
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 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
1) Find the area of the circleπ x r2
π x 42 π 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 = 503cm3
EXAMPLE- find the volume of this cylinder
10cm
4cm
QuestionsFind the volume of these cylinders, to 1 decimal place
1 2 3
654
4cm
12cm
3cm
10cm
5cm
15cm
3cm
18cm
7cm
14cm
2cm
11.3cm
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1 603.2
2 282.7
3 1178.1
ANSWERS
4 142.0
5 2155.1
6 508.9
Volume of Cylinders 2
1) Find the area of the circleπ x r2
π x 42 π 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 cylinder
Volume= 140cm3
4cm
h
Volume of Cylinders
1) Find the area of the circleπ 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 = 250Rearrange this to giver2 = 250 ÷ 94.2r2 =2.7 (1dp)r= 1.6 (1dp) cm
EXAMPLE- find the radius of this cylinder
Volume= 250cm3
r
30cm
QuestionsFind the volume of these cylinders, to 1 decimal place
1 2 3
654
4cm
h
3cm
h
5cm
h
r
8cm
r
14cm
r
12cm
volume= 100cm3volume= 120cm3volume= 320cm3
volume= 200cm3 volume= 150cm3volume= 90cm3
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ANSWERS1 6.42 4.23 1.34 2.35 1.86 1.9
Volume of Spheres
The formula for the volume of a sphere is
10cme.g
A= 4/3 x π x 103
A= 4/3 x π x 1000A=4188.8 cm3 (1 dp)
Volume of Cones
The formula for the volume of a cone is
10cme.g
A= 1/3 x π x 42 x 10A= 1/3 x π x 16 x 10A=167.6 cm3 (1 dp)
4cm
10cm
QuestionsFind the volume of these spheres, to 1 decimal place
1 2 3
654
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20cm 5cm
12cm
4cm
13cm
3cm
15cm
9cm
1 4188.82 33510.33 523.6
ANSWERS
4 201.15 122.56 1272.3
Circles Theorems
Angle at the centre
Angles connected by a
chord
Triangles made with a diameter
or radiiCyclic
Quadrilaterals
Tangents
50°
x
Example
Double AngleThe angle at the centre of a circle is twice the angle at the edge
Angle x = 50 x 2 x=100°
25°
x
160°100°
60°
135°
90°
xx
xxx
12 3
64 5
HOME
Answers1) 502)1203)1804)505)67.56)80
90°
Triangles inside circlesA triangle containing a diameter, will be a right angled triangle
A triangle containing two radii, will be isosceles
x
x
60°
x
1 2 3
31 2
72°
x
x x
x y
y
x
100°
x30°
22°
y
Answers1) X=302)x=183)x=454)X=40 y=405)x=30 y= 1206)x=22 y=136
x
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Angles connected by a chord
Angles connected by a chord are equal
x
x
y
y
25°x
12
3
645
y15°
yz
z
x
y
x
z
x
y
y
z
x
25° 53°30°
z
y
x
80°17°
95°35°
40°
125°
15°
40°
10°
100°
Answers1) x=25 y=152)x=125 y= 40 z=153)x=10 y=70 z=1004)X=105 y=40 z=355)x=53 y= 30 z=726)x=85 y=80 z=17
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90°
Tangents to a circleA tangent will always meet a radius at 90°
40°xy
z
3
120°
x
4
140°
x
2
x35 °
1
y
z
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Cyclic QuadrilateralsOpposite angles in a cyclic quadrilaterals add up to 180°
x
y
100° 60°
100 + y = 180 y=80°
60 + x = 180 x = 120 °
x
yx
y
x
y
95° 110° 54°
75°
20°
80°
x 2a
4b
15° 70° a
b
1
25°
yz
w
2 3
4 5Answers1) x=70 y=852)x=126 y=1053)x=100 y=1604)w=15 x=70 y=65 z= 255)a=60 b=36
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Here we will look at finding the area of sectors
You will need to be able to do two things:
Area of Segments
1) Find the area of a sector using the formula-
2) Find the area of a triangle using the formula-
Area= ½ absinC
Area of sector= Angle of Sector x πr2
360
C
b
a
Example-find the area of the blue segment
10cm10cm100°
Step 1- find the area of the whole sectorArea= 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 triangleArea= ½ absinC =1/2 x 10 x 10 x sin100 = 49.2cm2
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
12cm12cm120°
Step 1- find the area of the whole sectorArea= 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 triangleArea= ½ absinC =1/2 x 12 x 12 x sin120 = 62.4cm2
Step 3- take the area of the triangle from the area of the segment
150.8 – 62.4 = 88.4 cm2
Questions
10cm
130°11cm85°
12cm170°
5cm95°6.5cm
Find the area of the blue segments, to 1 decimal place
1 2 3
654
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17cm65°160°
ANSWERS1 75.12 29.53 201.14 8.35 51.86 33.0
Finding the Radius or angle of a Sector
r
Area= 100 x π x r2
360200= 100 x π x r2
360
200x360 = r2
100 x π
229.2=r2
15.1cm =r
Area=150
10cm x100° Area=200cm2
Area= θ x π x r2
360150= θ x π x 102
360
150x360 = θ102 x π
117.9°= θ
Questions
r
200°
r
175°
r
250°
5cmθ
6.5cm
θ17cm
θ
Find the missing radii and angles of these sectors, to 1 decimal place
1 2 3
654
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Area=100cm2 Area=120cm2Area=50cm2
Area=35cm2
Area=45cm2
Area=120cm2
ANSWERS1 7.62 8.33 5.44 160.45 122.16 47.6
The Equation of a CircleThe 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 CircleA 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 CircleA circle has the equation (x-5)2 + (y-7)2=16
What 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= 7±3.9y= 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:
a) (x-5)2 + (y-7)2=16
b) (x-3)2 + (y-8)2=36
c) (x+2)2 + (y-5)2=100
d) (x+2)2 + (y+5)2=49
e) (x-6)2 + (y+4)2=144
f) x2 + y2=4
g) x2 + (y+4)2=121
h) (x-1)2 + (y+14)2 -16=0
i) (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=6HOME
Answers1a) 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 -3.3
Answers2) 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 2
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
Here 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 = 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= 0Tidy 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 2Example 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= 0Tidy 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
QuestionsPut this equations into the form (x-a)2 + (y-b)2=r2 then find the centre and radius of the circle
1. x2 + y2 -8x – 4y- 5 =02. x2 + y2 -12x – 6y- 4 =0 3. x2 + y2 -4x – 10y- 20 =0 4. x2 + y2 -10x – 14y- 7 =0 5. x2 + y2 -12x – 2y- 62 =06. 2x2 +2y2 -20x – 20y- 28 =0 7. 3x2 + 3y2 -42x – 24y- 36 =0 8. 5x2 + 5y2 -100x – 30y- 60 =0
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Answers1) 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 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 bracketsx2-10x + 25 + 4x2 – 32x +64 = 16 simplify and make one side 05x2 -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 squareFactorising
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
0 solutions to the quadratic the circle and the line never meet
2 solutions to the quadratic
Intercepts between lines and circles
1) Find out whether these circles and lines intercept, if they do find the coordinates
of the interceptions
a) (x-5)2 + (y-7)2=16 and y=3x-1
b) (x-3)2 + (y-8)2=36 and y=2x-2
c) (x+2)2 + (y-5)2=100 and y=3x + 3
d) (x+2)2 + (y+5)2=49 and 2y+4=x
e) (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 Formulae
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