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Cambridge IGCSE0606

Circular measure

Mathematics Additional 0606 Chapter Notes

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Circular measure
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Circular measure notes

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In this chapter

  1. 1Radian measure
  2. 2Arc length
  3. 3Sector area
  4. 4Perimeter and the major arc
  5. 5Finding the radius and the angle
  6. 6Compound shapes and segments
  7. 7Exact answers and accuracy

1. Radian measure

In circular measure, angles are measured in radians. A full turn is 2π radians and a half turn is π radians.

A sector that is a fraction of the circle has that same fraction of 2π as its angle. Work in radians from the start: changing to degrees and back adds steps and loses accuracy.

full turn = 2π radians

half turn = π radians

Examiner insight

“Candidates should be reminded that it is not appropriate to use angles in degrees in circular measure questions. Answers obtained using degrees are likely to lose accuracy because of the extra steps involved.” “Some candidates appeared to be unclear with regard to radians with 1.75π being sometimes offered as the solution implying that some candidates thought it was a symbol for radians.”

Worked example 12 marks

Do not use a calculator. A sector is one sixth of a circle. Write down the angle of the sector in radians, in terms of π.

  1. 1

    a full turn is 2π radians [1]

  2. 2

    angle = 1/6 × 2π = π/3 radians [1]

2. Arc length

For a sector of radius r and angle θ radians, the arc length is rθ.

Use the formula directly. There is no π in it and no fraction of the circumference to find.

The angle must be in radians for this formula to work. On a calculator, check that the mode is set to radians before you start.

s = rθ

Examiner insight

“A simple use of the arc length was expected, an answer that could be written down with little or no working.” “A few candidates introduced a spurious factor of π into their formulae.”

Worked example 11 mark

In this question lengths are in centimetres and angles are in radians. A circle has centre C and radius 10. The points A and B lie on the circumference of the circle. Angle ACB is θ, where θ is 0.885 radians correct to 3 decimal places. Find the length of the arc AB.

  1. 1

    arc length = rθ = 10 × 0.885 = 8.85 cm [1]

3. Sector area

The area of a sector of radius r and angle θ radians is 1/2r²θ.

Halve r², not r. When the question gives an area in terms of π, write the equation first, then rearrange for the unknown. Keep the answer exact if the question asks for it.

A = 1/2r²θ

Worked example 14 marks

In this question lengths are in centimetres. A circle has centre O and radius r. Points A, B and C are on the circumference of the circle. The minor sector AOB has angle AOB = α radians, where α = π - 1. The area of the minor sector AOB is 18(π² - 1). Find the exact value of r. Simplify your answer.

  1. 1

    sector area: 1/2r²(π - 1) = 18(π² - 1) [1]

  2. 2

    r² = (36(π² - 1))/(π - 1) [1]

  3. 3

    π² - 1 = (π - 1)(π + 1), so r² = 36(π + 1) [1]

  4. 4

    r = 6√(π + 1) [1]

4. Perimeter and the major arc

The perimeter of a sector is the arc plus two radii. Do not stop at the arc.

The major arc is the rest of the circle. Its angle at the centre is 2π - θ, so its length is r(2π - θ). Sketch the shape and label every length before you add.

P = 2r + rθ

major arc = r(2π - θ)

Examiner insight

“A few candidates equated the arc length to the perimeter and would possibly have benefitted from making a sketch.” “There was some confusion between major and minor arcs.”

Worked example 14 marks

In this question lengths are in centimetres. A circle has centre O and radius r. Points A, B and C are on the circumference of the circle. The minor sector AOB has angle AOB = α radians. The perimeter of the minor sector AOB is equal to the length of the major arc, ACB. Find the exact value of α.

  1. 1

    perimeter of the sector = 2r + rα [1]

  2. 2

    major arc = r(2π - α) [1]

  3. 3

    2r + rα = r(2π - α), so 2 + α = 2π - α [1]

  4. 4

    2α = 2π - 2, so α = π - 1 [1]

5. Finding the radius and the angle

A question may give two facts about one sector, such as its arc length and its area. Write one equation for each, then solve together.

Since s = rθ, the area is also 1/2rs. An angle of 1 radian has an arc equal to the radius, so an arc greater than the radius means an angle greater than 1 radian.

θ = s/r

A = 1/2rs

Examiner insight

“This question was very well answered by candidates who worked in radians, but those who attempted to use degrees rarely made progress.” “The simplest explanation was to state that the arc length was greater than the radius.”

Worked example 13 marks

A sector has arc length 12 cm and area 30 cm². Find the radius of the sector and its angle in radians.

  1. 1

    rθ = 12 and 1/2r²θ = 30 [1]

  2. 2

    1/2 × r × 12 = 30, so r = 5 cm [1]

  3. 3

    θ = 12 ÷ 5 = 2.4 radians [1]

6. Compound shapes and segments

A triangle with two sides r and the angle θ between them has area 1/2r²sinθ. A segment is the sector minus the triangle.

For a compound shape, split it into sectors, triangles and segments. State the plan first: which areas you add and which you subtract. Then calculate.

triangle = 1/2r²sinθ

segment = 1/2r²θ - 1/2r²sinθ

Examiner insight

“Most errors were due to an incorrect plan.” “It was often unclear what candidates intended to add to the sector area with many adding both the area of triangle AOB and the area of the kite AOBC.”

Worked example 15 marks

A badge is made from an equilateral triangle ABC with side x cm. The arcs BC, CA and AB are parts of the circles with centres A, B and C respectively. The badge is the region enclosed by the three arcs. The area of the badge is 2(π - √3) cm². Find x.

  1. 1

    each angle of the triangle is π/3, so sector area = 1/2x² × π/3 [1]

  2. 2

    triangle area = 1/2x² sinπ/3 = 1/2x² × (√3)/2 = (√3x²)/4 [1]

  3. 3

    badge = 3 × sector - 2 × triangle [1]

  4. 4

    = 3 × (π x²)/6 - 2 × (√3x²)/4 = x²/2(π - √3) [1]

  5. 5

    x²/2(π - √3) = 2(π - √3), so x² = 4 and x = 2 [1]

7. Exact answers and accuracy

When a question asks for an exact value, leave π and surds in the answer. A rounded decimal does not earn the mark.

When a question says show that, work to more places than the value you are given, state that longer value, then round it. Otherwise keep the working exact and round only the final value.

Examiner insight

“It is better to keep calculations in exact form for method and only round to 3 significant figures for the final value.” “To obtain full marks, it was essential that a statement of angle AOB = 1.696 (or greater accuracy) so angle AOB = 1.70 to 2 decimal places, or equivalent wording, was seen.”

Worked example 16 marks

In this question lengths are in centimetres and angles are in radians. A circle has centre C and radius 10. The points A and B lie on the circumference of the circle. The point P is such that CBP is a straight line. The line BP has length x. The line AP is a tangent to the circle at A. Angle ACB is θ. The perimeter of the triangle APC is 38. Show that the value of θ is 0.885 radians correct to 3 decimal places.

  1. 1

    CP = 10 + x and AP² = (10 + x)² - 10² = 20x + x² [1]

  2. 2

    perimeter: 10 + 10 + x + AP = 38, so AP = 18 - x [1]

  3. 3

    (18 - x)² = 20x + x² [1]

  4. 4

    324 - 36x = 20x, so x = 324/56 = 5.7857… [1]

  5. 5

    cosθ = 10/(10 +x) = 10/(15.7857…) [1]

  6. 6

    θ = 0.8847… = 0.885 to 3 decimal places [1]

Recap

  • A full turn is 2π radians. Work in radians, not degrees.
  • Arc length s = rθ.
  • Sector area A = 1/2r²θ.
  • Perimeter of a sector = 2r + rθ. Major arc = r(2π - θ).
  • Two facts about one sector give two equations: solve them together.
  • Segment = sector minus triangle. State the plan before calculating.
  • Keep π for an exact value. For show that, state the longer value, then round.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write down the number of radians in a half turn.1 mark
  2. Do not use a calculator. A sector has radius 6 cm and angle π/3 radians. Find the exact length of its arc.2 marks
  3. A sector has radius 8 cm and angle 1.4 radians. Find its area.2 marks
  4. A sector has radius 5 cm and angle 2.2 radians. Find its perimeter.3 marks
  5. Do not use a calculator. A sector of a circle has radius 12 cm and arc length 8π cm. Find the angle of the sector in radians, in terms of π.3 marks
  6. A sector has angle 0.75 radians and area 24 cm². Find its radius.3 marks
  7. Do not use a calculator. A circle has radius 9 cm. A sector of the circle has angle 2π/3 radians. Find the exact length of the major arc, which is the rest of the circumference.3 marks
  8. Do not use a calculator. A flower bed is the region between a chord and the minor arc of a circle of radius 6 m. The chord joins two points on the circle and makes an angle of π/3 radians at the centre of the circle. Find the exact area of the bed and the exact perimeter of the bed.6 marks
  9. Do not use a calculator. A piece of wire 40 cm long is bent to make the outline of a sector of a circle. The sector has radius r cm, angle θ radians and area 36 cm². Find the value of r and the value of θ, explaining why there is only one possible value of r.6 marks
  10. A circle has centre O and radius 8 cm. The point P is 17 cm from O. The two tangents from P touch the circle at A and B. A metal sign is the region outside the circle enclosed by the lines PA and PB and the minor arc AB. Find the perimeter of the sign and the area of the sign.8 marks

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