Unformatted text preview: The Circle
General Main Parts of a Circle www.mathsrevision.com Investigation of the Ratio of Circle Circumference of the circle Composite shapes Perimeter Diameter = Circumference Area of a circle Composite Area Starter Questions
General 1. W hat shape has an inf int e line symmet ry. www.mathsrevision.com 2. Solve 3x+7 28. 3. Find t he perimet er of t he st ar. 4. Convert your answer t o Q3 t o (m). 7cm Monday 13 February 2012 Created by Mr Lafferty 2 Main Parts of a Circle
General Learning Intention Success Criteria 1. Know the terms circumference, diameter 2. 3. using Identify them on a Calculate the formula. www.mathsrevision.com 1. To revise the basics of the circle. and radius. circle. circumference 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Main part of a Circle
General www.mathsrevision.com Main parts of the circle
1 r = D 2 O D = 2r
Monday 13 February 2012 Created by Mr Lafferty 4 Main part of a Circle
General www.mathsrevision.com 2cm cm 10 1 r = D r = 5c m 2
Monday 13 February 2012 Created by Mr Lafferty D = 2r D = 4c m 5 Main part of a Circle
General www.mathsrevision.com Now try Exercise 1 Ch9 (page 100) 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Starter Questions
General 1. The diamet er of a circle is 10cm. W hat is t he radius. www.mathsrevision.com 2. The radius of a circle is 3. 5cm. W hat is t he diamet er. 3. Find t he perimet er of t he st ar. 4. Convert your answer t o Q3 t o (mm). 5cm Monday 13 February 2012 Created by Mr Lafferty 7 Stars In Your Eyes
General "Today Matthew we are going to be" www.mathsrevision.com Archimedes of Syracuse Monday 13 February 2012 Created by Mr Lafferty 8 History of Circles
General 75 years old www.mathsrevision.com 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 steps and arrive at the equation for determining the circumference for any given circle. Monday 13 February 2012 Created by Mr Lafferty 9 Parts of the Circle
Circumference General
The radius is measured from the centre of the circle to the edge. www.mathsrevision.com O radius
O = centre of circle The diameter is measured from one edge to the other passing through the centre of the circle. diameter Monday 13 February 2012 Created by Mr Lafferty 10 Circle Investigation
General Construct a table shown below to enable us to record our results. www.mathsrevision.com Circle I nvest igat ion
Circle 1 2 3 4 5 6 7 8 Circumf erence = C Diamet er = D C+D= Calculat ions
C D= C/ D= CxD= Monday 13 February 2012 Created by Mr Lafferty 11 Investigation of the Circle
General Our Investigation
O www.mathsrevision.com To find a relationship between the diameter and circumference of a given circle. How can we measure the diameter and circumference
Monday 13 February 2012 Created by Mr Lafferty Question ? 12 Measuring the Diameter General of a circle
O
O www.mathsrevision.com Use a ruler The diameter is the largest distance between one side of a circle to the other passing through the centre O.
Monday 13 February 2012 Created by Mr Lafferty 13 Measuring the Circumference General of a circle www.mathsrevision.com O Flexible tape measure Take a tape measure and put it round the circle Read off the measurement
Monday 13 February 2012 Created by Mr Lafferty 14 Measuring the Circumference of a circle
General Roll along an even surface www.mathsrevision.com Starting point End point One complete rotation equals the length of the circumference Be careful to avoid slip!
Monday 13 February 2012 Created by Mr Lafferty 15 Circle Investigation
General
Circle I nvest igat ion
Circle 1
2 3 3 Cir cumf Circumf erence = C Calculat ions
C+D= 23. 1 20. 9 18. 8 16. 6 14. 7 12. 5 10. 5 8. 4 C D= 12. 1 10. 9 9. 8 8. 6 7. 7 6. 5 5. 5 4. 4 C/ D= 3. 2 3. 2 3. 2 3. 2 3. 2 3. 2 3. 2 3. 2 CxD= 96. 8 79. 5 64. 4 50. 4 39. 2 28. 5 20. 0 12. 8 5. 5
5. 0 Diamet er = D 17. 6
15. 9 14. 3
12. 6 4. 5
4. 0 4 5 5
6 6 7 7 8 8 11. 2
9. 5 9. 5 8. 0 8. 0 6. 4 6. 4 3. 5
3. 0 3. 0 2. 5 2. 5 2. 0 2. 0 Monday 13 February 2012 Created by Mr Lafferty 16 Circle Investigation
General www.mathsrevision.com Us ing y o ur re s ults write d o wn, in yo ur o wn wo rd s , a n a p p ro xim a te re la tio ns h ip b e twe e n th e c irc um fe re nc e a nd d ia m e te r fo r a g ive n c irc le . "C irc um fe re nc e a p p ro xim a te ly e q ua ls th re e a nd b it d ia m e te rs " Ac tua l va lue is 3 .1 4 wh ic h we write a s Pronounced C =D "Pi" C = circumference
D = diameter Monday 13 February 2012 Created by Mr Lafferty ; 3.14 17 Circle Investigation
General www.mathsrevision.com No Exercise 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Starter Questions
General 1. N ame t his shape. www.mathsrevision.com 4
2 . C a lc ula t e ( 3 m ) x ( 4 m ) 3. Find t he mean, median and mode of dat a set
13 Feb 2012 3, 5, 5, 7, 10
Created by Mr. Lafferty Maths Dept. Main Parts of a Circle
General Learning Intention Success Criteria 1. Know the terms circumference, diameter 2. 3. using Identify them on a Calculate the formula. www.mathsrevision.com 1. To revise the basics of the circle. and radius. circle. circumference 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Main Parts of a Circle
General Main parts of the circle
www.mathsrevision.com
1 r = D 2 O C = D
13 Feb 2012 Circumference D = 2r
Created by Mr. Lafferty Maths Dept. General Calculating the Circumference
Example : Find the length of the circumference (Perimeter) of each circle www.mathsrevision.com 2cm m 0c 1 C = D = 10
Created by Mr. Lafferty Maths Dept. D = 2r D = 4c m = 3 1.4 c m
13 Feb 2012 C = D = = 12 .5 6 c m 4 General Calculating the Circumference
Q. Calculate the curved part of this shape.
Solution www.mathsrevision.com 6m
90
o C = D 1 c ur ve d le ng t h = D 4 1 c ur ve d le ng t h = 12 4 c ur ve d le ng t h = 3 Created by Mr Lafferty 23 c ur ve d le ng t h = 9 .4 m
Monday 13 February 2012 Main part of a Circle
General www.mathsrevision.com Now try Exercise 2 Ch9 (page 100) 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Starter Questions
General 1. Find t he cir cumf er ence of t he cir cle.
4 www.mathsrevision.com 2. 3.
13 Feb 2012 Calculat e ( 7 ) x ( 12 ) 6 Find t he mean, median and mode of dat a set 6, 7, 3, 10, 8, 10, 12
Created by Mr. Lafferty Maths Dept. Composite Perimeter
General Learning Intention Success Criteria 1. Recall knowledge of circles so far. 2. Solve mixed problems by applying all our knowledge so far. www.mathsrevision.com 1. To give some examples of problems that we can solve by applying our knowledge of circles and of the course so far. 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Composite Perimeter
General Things to think about when doing exercise. www.mathsrevision.com 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 !
13 Feb 2012 Created by Mr. Lafferty Maths Dept. Composite Perimeter
General Q. Find the perimeter for the semicircle shape ? www.mathsrevision.com 1 P = D + 2 0 2
180
o Solution 1 P= 2 20 +20 P = 10 + 2 0
Monday 13 February 2012 Created by Mr Lafferty P = 5 1.4 c m 28 Composite Perimeter
General Q. Find the perimeter of the shape below ? www.mathsrevision.com
8cm
90
o 1 P = D + 8 + 8 4 1 P = 16 + 8 + 8 4 P = 4 + 16 P = 2 8 .6 c m Solution Monday 13 February 2012 Created by Mr Lafferty 29 Composite Perimeter
General Example 1 : Find the perimeter of this shape Perimeter = 3 sides + semicircle www.mathsrevision.com 1 P = 5 + 20 + 5 + 2
5 cm 20 P = 30 + 10 P = 61.4cm
20cm
13 Feb 2012 Created by Mr. Lafferty Maths Dept. Composite Perimeter
General www.mathsrevision.com Now try Extension Booklet 6E (page 125) 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Starter Questions
General 1. N ame t he shape. www.mathsrevision.com 2 . C a lc ula t e ( 5 d ) x ( 6 d ) 3. Find t he mean, median and mode of dat a set
13 Feb 2012 3, 5, 6, 7, 9
Created by Mr. Lafferty Maths Dept. Finding the Diameter
General The Circle Learning Intention Success Criteria 1. Understand how to rearrange circumference formula to find diameter. 2. Solve diameter problems. www.mathsrevision.com 1. To explain how we can find diameter of a circle if we know the circumference. 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Finding the Diameter
General The Circle www.mathsrevision.com We can easily rearrange the circumference formula so that we have the diameter D on one side. C = D
C D= 13 Feb 2012 Created by Mr. Lafferty Maths Dept. You have 1 minute to rearrange equation. Remember change side change sign Finding the Diameter
General Example : Find the diameter of each circle given the circumference.
C = 62.8 cm C =15.7 cm The Circle www.mathsrevision.com m 5c 20cm 13 Feb 2012 C 6 2 .8 D= = = 2 0 c m C 15 .7 D= = = 5 c m
Created by Mr. Lafferty Maths Dept. Finding the Diameter
General www.mathsrevision.com Now try Extension Booklet 4E (page 32) 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Starter Questions
General 1.
www.mathsrevision.com N ame t his shape. 2 . C a lc ula t e ( 5 b ) x ( 4 b ) 3.
13 Feb 2012 2x  3 = 29
Created by Mr. Lafferty Maths Dept. Peeling an orange General A x Area of a Circle
circumference A x 2 r B www.mathsrevision.com r
O O What do we call the distance OA in terms of the large circle. radius (r ) circle. What do we call the distance AB in terms of circumference (2 r ) the large Area of a Circle
General A 2 r
x B www.mathsrevision.com r
O What is the formula for the area of a rightangle triangle. 1 A= base height 2
Use this formula to work out the area for a circle. 1 A= 2 r 2r A = r2
13 Feb 2012 Created by Mr. Lafferty Maths Dept. Area of a Circle
General www.mathsrevision.com 8 7 6 5 1 2 3 4 If we break the circle into equal sectors And lay them out side by side We get very close to a rectangle. 1 2 3 4 5 6 7 8 Monday 13 February 2012 Created by Mr Lafferty 40 Area of a Circle
General www.mathsrevision.com 1 2 3 4 5 6 7 8 thinner and thinner sectors 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
Monday 13 February 2012 Created by Mr Lafferty 41 Area of a Circle
General www.mathsrevision.com r r
A r e a o f a r e c t ang le = l A r e a o f a r e c t ang le = r b r = r
2
42 2 But the area inside this rectangle is also the area of the circle A r e a o f a c ir c le = r
Monday 13 February 2012 Created by Mr Lafferty Area of a circle
General Q. Find the area of the circle ?
Solution 4cm www.mathsrevision.com A = r
A = 2 4 2
2 A = 5 0 .2 6 c m
13 Feb 2012 Created by Mr. Lafferty Maths Dept. S3 The Area of a circle
Q. The area of a circle is 12.64 cm
2 www.mathsrevision.com Find its radius? .
Solution A = r 12 .6 4 = 2 r 2 r 2 12 .6 4 = = 4c m r = 4 = 2c m
Monday 13 February 2012 Created by Mr Lafferty 44 Area of a circle
General Q. The diameter of the circle is 60cm. Find area of the circle? www.mathsrevision.com Solution D 60 r = = = 3 0c m 2 2 A = r 2 A = 30
13 Feb 2012 Created by Mr. Lafferty Maths Dept. 2
2 A = 2 8 2 6 c m Area of a Circle
General www.mathsrevision.com Now try Extension booklet 5E (page 35) 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Starter Questions
General 1. Find t he lengt h of t he cur ved par t . www.mathsrevision.com 2cm 2. 3.
13 Feb 2012 Calculat e 2 0 + 8 Simplif y 4 3(5y  2)
Created by Mr. Lafferty Maths Dept. Composite Area
General Learning Intention Success Criteria 1. Recall knowledge of circles so far. 2. Solve mixed problems by applying all our knowledge so far. www.mathsrevision.com 1. To give some examples of problems that we can solve by applying our knowledge of circles and of the course so far. 13 Feb 2012 Created by Mr. Lafferty Maths Dept. Composite Area
General Example 1 : Find the area of the shape Area = rectangle + semicircle www.mathsrevision.com 1 2 A=l b + r 2
5 cm 20cm
13 Feb 2012 1 2 A = 20 5 + ( 10) 2 A = 100 + 157 A = 257 cm 2
Created by Mr. Lafferty Maths Dept. Composite Area
General Example 1 : Find the area of the red part. www.mathsrevision.com Area = Big Circle Small Circle
4cm A = r 2 B
2  r 2 S A= 5  A = 21
A = 66 cm 2
Created by Mr. Lafferty Maths Dept. 2 2 10cm
13 Feb 2012 Composite Area
General Example 2 : A circle is contained in a square. Find the grey shaded area below.
Area = square circle www.mathsrevision.com 8cm A=l 2 r 2 A = 8 8 ( 4) 2 A = 64  50.24 A = 13.76 cm 2
13 Feb 2012 Created by Mr. Lafferty Maths Dept. Composite Area
General www.mathsrevision.com Now try Extension booklet 6E (page 38) 13 Feb 2012 Created by Mr. Lafferty Maths Dept. ...
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