Essential Knowledge Recap Essential Knowledge Recap. Calculate the missing sides of these triangles. 30° 8 30° ? 8 Check ability to calculate missing sides using sin. ?

2000 years ago Archimedes asked: for the most basic circle, with diameter 1, what is the perimeter? What would you estimate the perimeter to be? If this diameter is 1, do you think the perimeter is more than 1? Less? How many times more?

1 We call the perimeter of this, the most basic circle, Pi (π) Archimedes knew that π was less than 4. How? 1 He drew a square around the outside of the circle. We can see that this will be more than the length of the perimeter of the circle

1 Why must the perimeter of the outside square be exactly 4? The diameter is 1, this means the height of the circle is also 1. So all sides of the square are 1.

1 He then drew a square inside the circle. What is the perimeter of this square? 1 Using pythagoras, the sides of each triangle are 0.5. Square root of (0.5^2 + 0.5^2) = 0.707 *4 = 2.83 rounded

1 So Pi (π) is between 2.83 and 4 Can we get closer? How could we make a better estimate of the perimeter?

1 This shape is closer to the perimeter of the basic circle. How can we work out its perimeter? 1 We can split any polygon into triangles…

1 What do we know about these triangles? We know they are isosceles and the sides are 0.5. We also know that the angle at the centre is 360/8 = 45 degrees

1 We can work out the opposite side using trigonometry. The opposite side is sin(22.5) *0.5 = 0.19…. There are 16 of these that make up the perimeter of the octagon. 0.19*16 = 3.06…

1 So we now know that Pi is between 3.06 and 4. Can we be more accurate? 1 The opposite side is sin(22.5) *0.5 = 0.19…. There are 16 of these that make up the perimeter of the octagon. 0.19*16 = 3.06…

Determine the perimeter when the circle is split into 10. What about 20? 100? Split into 10: Angle for each part is 36 degrees. Sin(36/2)*0.5*20 = 3.09 Split into 20: Angle is 18 degrees. Sin(18/2)*0.5*40 = 3.128 Split into 100: Angle is 36/100 = 3.6 degrees. Sin(3.6/2)*0.5*200 = 3.141…

1 In Summary The most basic circle has a dimeter of 1. We call its perimeter Pi (π) Pi can never be written down. We will never be able to work out all its digits because we would need to split the circle into an infinite number of parts. So Pi goes on forever. It starts 3.14… Numbers like this, that continue forever, are called irrational numbers. 1

2 1 What is the perimeter of a circle with diameter 2? This is an enlargement. All the lengths get twice as long. Therefore, the perimeter is 2π, or we can use our calculator to estimate it as a decimal.

d 1 What is the perimeter of a circle with diameter d? The perimeter of any circle, called the circumference, is pi times diameter.

The circumference of any circle is Pi (π) × diameter In Summary The special name for the perimeter of a whole circle is circumference. The circumference of any circle is Pi (π) × diameter (C = πd)

Calculate the perimeter of each of these shapes. 3 2 3

Extension If the whole circle has a diameter of 1, what is the perimeter of the black section?

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