11: :00 Circles and Pi 1

What is a circle? Help pupils articulate it:  Draw what they describe  Encourage pupils to improve description if stuck,  Encourage pupils to describe ‘what you do’ to get a circle?  How many sides does it have? 2

diameter Circumference radius chord Major Segment Minor Segment Minor Arc Major Arc Minor Sector Major Sector Name parts of a circle you remember/know. Tangent 3

Give out circles Locate the centre of the circle given to you. Be as accurate as you can. Use whatever method you like (NO COMPASSES THOUGH) When you have located the centre, then use compasses to draw a series of concentric circles. Does it look right?  How many different exact methods can you find for locating the centre of the circle?  What if the circles were made of plastic? 4

Assuming that you do not know any formulae about areas/perimeters of circles,  How would you find out the circumference of the tin can in front of you?  How about the diameter.  Collect data from all. Is there a relationship between the circumference and the diameter of the tins/cans? 5

GSP 6

The circumference of a circle For any circle, π = circumference diameter or, We can rearrange this to make an formula to find the circumference of a circle given its diameter. C = πd π = C d 7

π Make a list of facts about pi. Approximations of pi  8

π =

Irrational number Position on a number line 10

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Transform The Area of a Circle 4 Sectors 12

8 Sectors Transform The Area of a Circle 13

32 Sectors Transform Remember C = 2 π r ? ? As the number of sectors  , the transformed shape becomes more and more like a rectangle. What will the dimensions eventually become? ½C r πr A = πr x r = πr 2 14

Measuring angles in degrees The use of the number 360 is thought to originate from the approximate number of days in a year. The system of using degrees to measure angles, where 1° is equal to of a full turn, is attributed to the ancient Babylonians. 360 is also a number that has a high number of factors and so many fractions of a full turn can be written as a whole number of degrees. An angle is a measure of rotation. For example, of a full turn is equal to 160°. 15

Measuring angles in radians A full turn is divided into 2 π radians. One radian is therefore equal to the angle subtended by an arc of length r. In many mathematical and scientific applications, particularly in calculus, it is more appropriate to measure angles in radians. Remember that the circumference of a circle of radius r is equal to 2 πr. 1 radian can be written as 1 rad or 1 c. 2 π rad = 360° So: 1 rad = r r r 1 rad O 16

Converting radians to degrees We can convert radians to degrees using: Radians are usually expressed as fractions or multiples of π so, for example: 2 π rad = 360° Or: π rad = 180° If the angle is not given in terms of π, when using a calculator for example, it can be converted to degrees by multiplying by For example: 17