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KS4 Mathematics

KS4 Mathematics. S5 Circles. S5 Circles. Contents. S5.1 Naming circle parts. A. S5.2 Angles in a circle. A. S5.4 Circumference and arc length. S5.3 Tangents and chords. A. A. S5.5 Areas of circles and sectors. A. The value of π. π = 3.141592653589793238462643383279502884197169

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KS4 Mathematics

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  1. KS4 Mathematics S5 Circles

  2. S5 Circles Contents S5.1 Naming circle parts • A S5.2 Angles in a circle • A S5.4 Circumference and arc length S5.3 Tangents and chords • A • A S5.5 Areas of circles and sectors • A

  3. The value of π π = 3.141592653589793238462643383279502884197169 39937510582097494459230781640628620899862803482 53421170679821480865132823066470938446095505822 31725359408128481117450284102701938521105559644 62294895493038196 (to 200 decimal places)! For any circle the circumference is always just over three times bigger than the radius. The exact number is called π (pi). We use the symbol π because the number cannot be written exactly.

  4. Approximations for the value of π Better approximations are 3.14 or . 22 7 When we are doing calculations involving the value π we have to use an approximation for the value. For a rough approximation we can use 3. We can also use the π button on a calculator. Most questions will tell you what approximations to use. When a calculation has lots of steps we write π as a symbol throughout and evaluate it at the end, if necessary.

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

  6. The circumference of a circle Use π = 3.14 to find the circumference of this circle. C = πd 9.5 cm = 3.14 × 9.5 = 29.83 cm

  7. Finding the circumference given the radius The diameter of a circle is two times its radius, or d = 2r We can substitute this into the formula C = πd to give us a formula to find the circumference of a circle given its radius. C = 2πr

  8. The circumference of a circle 9 m 4 cm 58 cm 23 mm Use π = 3.14 to find the circumference of the following circles: C = πd C = 2πr = 3.14 × 4 = 2 × 3.14 × 9 = 12.56 cm = 56.52 m C = πd C = 2πr = 3.14 × 23 = 2 × 3.14 × 58 = 72.22 mm = 364.24 cm

  9. Finding the radius given the circumference C = 2π 12 2 × 3.14 Use π = 3.14 to find the radius of this circle. C = 2πr 12 cm How can we rearrange this to make r the subject of the formula? r = ? = 1.91 cm (to 2 d.p.)

  10. Finding the length of an arc A 6 cm B What is the length of arc AB? An arc is a section of the circumference. The length of arc AB is a fraction of the length of the circumference. To work out what fraction of the circumference it is we look at the angle at the centre. In this example, we have a 90° angle at the centre.

  11. Finding the length of an arc A The arc length is of the circumference of the circle. 6 cm 1 90° = 1 1 1 4 4 4 4 360° B Length of arc AB = × 2πr × 2π× 6 = What is the length of arc AB? This is because, So, Length of arc AB = 9.42 cm (to 2 d.p.)

  12. Finding the length of an arc θ 360 Arc length AB = × 2πr 2πrθ πrθ Arc length AB = = 360 180 A B r θ For any circle with radius r and angle at the centreθ, This is the circumference of the circle.

  13. Finding the length of an arc

  14. The perimeter of shapes made from arcs 40° 1 = 360° 9 1 1 × π× 6 + × π× 12 + 2 9 1 1 × π× 4 + × π× 6 + 9 2 1 × π× 2 2 Find the perimeter of these shapes on a cm square grid: 40° The perimeter of this shape is made from three semi-circles. Perimeter = Perimeter = 3 + 3 = 6πcm = 2π+ 6 = 18.85cm (to 2 d.p.) = 12.28cm (to 2 d.p.)

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