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Cambridge IGCSE0607

Circles, arcs and sectors

International Mathematics 0607 Chapter Notes

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Circles, arcs and sectors
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In this chapter

  1. 1Circumference and area of a circle
  2. 2Finding the radius from the circumference
  3. 3Arc length and sector area
  4. 4Perimeter of a sector and of shapes made from arcs
  5. 5Minor and major sectors
  6. 6Finding the radius from a sector

1. Circumference and area of a circle

The circumference is the distance round a circle. The area is the space inside it. Both use the radius, which is half the diameter.

When a question gives the diameter, halve it first. When it says in terms of π, keep π in the answer and do not multiply it out.

C = 2π r = π d

A = π r²

Examiner insight

“First, candidates had to pick the correct formula for the area of a circle but some used that for the circumference.” “Next, candidates had to substitute the value of the radius, not the diameter, into the correct formula.”

Worked example 13 marks

The circumference of a circle is 12π metres.

Work out the area of the circle. Give your answer in terms of π.

  1. 1

    2π r = 12π [1]

  2. 2

    r = 6 [1]

  3. 3

    A = π × 6² = 36π m² [1]

2. Finding the radius from the circumference

To go from the circumference to the area, find the radius first: divide the circumference by 2π.

On a graphic display calculator, enter the division with brackets, 58 ÷ (2 × π), and store the result. Use the stored value in π r², so no rounding happens early. Round only the final answer, to three significant figures.

r = C/2π

Examiner insight

“Some candidates mistook the circumference for the diameter and halved this value for their radius.” “Some candidates worked out the radius to only two significant figures and this affected the accuracy of their answer for the area.”

Worked example 13 marks

A circle has circumference 58 cm.

Work out the area of the circle.

  1. 1

    r = 58/2π = 9.2309… [1]

  2. 2

    A = π × 9.2309…² [1]

  3. 3

    = 268 cm² [1]

3. Arc length and sector area

A sector is the region between two radii and an arc. Its angle θ at the centre makes it a fraction θ/360 of the whole circle.

The arc length is that fraction of the circumference. The sector area is that fraction of the area. Use the circumference for a length and the area for an area.

arc length = θ/360 × 2π r

sector area = θ/360 × π r²

Examiner insight

“The most common error was to use a formula for area rather than for arc length.” “Only a handful of candidates knew how to calculate the area of the sector.”

Worked example 14 marks

Do not use a calculator.

A sector of a circle has radius 12 cm and sector angle 90°.

(a) Find the arc length. Give your answer in terms of π.

(b) Find the area of the sector. Give your answer in terms of π.

  1. 1

    (a) fraction = 90/360 = 1/4 [1]

  2. 2

    arc length = 1/4 × 2π × 12 = 6π cm [1]

  3. 3

    (b) sector area = 1/4 × π × 12² [1]

  4. 4

    = 36π cm² [1]

4. Perimeter of a sector and of shapes made from arcs

The perimeter of a sector is the arc plus both radii. A quarter circle has a perimeter of one quarter of the circumference plus two radii.

For a shape between two sectors, add the two arc lengths and then add the straight edges. Keep the answer in terms of π and write it as a + bπ.

P = θ/360 × 2π r + 2r

Examiner insight

“Most candidates realised that they needed to add on 6 + 6 to their total arc lengths.” “Many candidates decided to subtract their arc lengths due to being confused with similar problems that required the area of the sector.”

Worked example 14 marks

OAD is a sector of a circle with centre O and sector angle 120°. OBC is a sector of a circle with centre O and sector angle 120°. ABO and DCO are straight lines. OA = 9 cm and OB = 3 cm.

The shaded region ABCD is the part of sector OAD outside sector OBC. The perimeter of the shaded region ABCD is (a + bπ) cm.

Find the value of a and the value of b.

  1. 1

    arc AD = 120/360 × 2π × 9 = 6π [1]

  2. 2

    arc BC = 120/360 × 2π × 3 = 2π [1]

  3. 3

    AB = DC = 9 - 3 = 6 [1]

  4. 4

    perimeter = 6π + 2π + 6 + 6 = 12 + 8π, so a = 12 and b = 8 [1]

5. Minor and major sectors

A minor sector has an angle less than 180°. A major sector has an angle greater than 180°. The matching arcs are the minor arc and the major arc.

If a question gives the minor angle and asks for the major sector, use 360 - θ as the angle.

Examiner insight

“Almost all candidates knew how to find the length of an arc but a substantial number found the length of the minor arc.” “Many just tried to find the length of the chord SR suggesting that the term minor arc was unfamiliar.”

Worked example 15 marks

Do not use a calculator.

A circle has radius 6 cm. A minor sector of the circle has angle 120°.

(a) Find the length of the major arc. Give your answer in terms of π.

(b) Find the area of the major sector. Give your answer in terms of π.

  1. 1

    major angle = 360 - 120 = 240^(°) [1]

  2. 2

    (a) 240/360 × 2π × 6 [1]

  3. 3

    = 8π cm [1]

  4. 4

    (b) 240/360 × π × 6² [1]

  5. 5

    = 24π cm² [1]

6. Finding the radius from a sector

When the area of a sector is given, write the sector formula as an equation and solve for r. Rearrange for r² first, then take the square root.

θ/360 × π r² = A ⇒ r = √(360A/πθ)

Worked example 13 marks

A sector of a circle has radius r cm and sector angle 80°. The area of the sector is 62.6 cm².

Calculate the value of r.

  1. 1

    80/360 × π r² = 62.6 [1]

  2. 2

    r² = (62.6 × 360)/80π = 89.67… [1]

  3. 3

    r = 9.47 [1]

Recap

  • Circumference = 2π r. Area = π r². The radius is half the diameter.
  • In terms of π means keep π in the answer.
  • Radius from circumference: r = C ÷ 2π. Store it unrounded.
  • Arc length = θ/360 of the circumference. Sector area = θ/360 of the area.
  • Perimeter of a sector = arc + two radii.
  • Extended: a major sector has an angle over 180°. Use 360 - θ if you are given the minor angle.
  • Extended: for the radius of a sector, solve for r² then take the square root.

End-of-chapter exercise

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

  1. Do not use a calculator. A circle has diameter 14 cm. Find its circumference. Give your answer in terms of π.2 marks
  2. Do not use a calculator. A circle has diameter 10 cm. Find its area. Give your answer in terms of π.2 marks
  3. Do not use a calculator. A circle has area 49π cm². Find the circumference of the circle. Give your answer in terms of π.3 marks
  4. A circle has circumference 44 cm. Calculate the radius and the area of the circle. Give each answer correct to 3 significant figures.3 marks
  5. Do not use a calculator. A sector has radius 12 cm and sector angle 60°. Find the arc length. Give your answer in terms of π.2 marks
  6. Do not use a calculator. A sector has radius 6 cm and sector angle 60°. Find the area of the sector. Give your answer in terms of π.2 marks
  7. A sector has radius 8.5 cm and sector angle 40°. Calculate the perimeter of the sector. Give your answer correct to 3 significant figures.3 marks
  8. Do not use a calculator. A circle has radius 12 cm. A minor sector of the circle has angle 60°. Find the length of the major arc. Give your answer in terms of π.3 marks
  9. Do not use a calculator. A windscreen wiper arm turns about a fixed point O. The rubber blade covers the part of the arm from 10 cm to 50 cm from O. The arm turns through an angle of 135°, and the blade cleans the region of glass it passes over. (a) Find the area of glass cleaned. Give your answer in terms of π. (b) The perimeter of the cleaned region is (a + bπ) cm. Find the value of a and the value of b.5 marks
  10. A circular pond has a wooden deck built over part of it. The deck is a minor sector of the circle with sector angle 100°. The water surface that is left is the major sector, and its area is 95 m². A rope is laid all the way round the edge of the water surface. Calculate the length of the rope. Give your answer correct to 3 significant figures.6 marks

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