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Cambridge IGCSE0580

Circles, arcs and sectors

Mathematics 0580 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. 2Working back to the radius
  3. 3Arc length and sector area
  4. 4Perimeter of a semicircle or a sector
  5. 5Minor and major sectors

1. Circumference and area of a circle

The circumference is the distance round a circle. The area is the space inside it. Both formulas are in the List of formulas, so check which one you need.

The formulas use the radius, which is half the diameter. Halve the diameter before you use the area formula.

If the question says in terms of π, leave π in the answer. Do not change it to a decimal.

C = 2π r = π d

A = π r²

Examiner insight

“Many worked out a value for the area and even those who did find the correct answer in terms of π often changed it to a numerical value for their final answer.”

Worked example 14 marks

Do not use a calculator. A circle has diameter 10 cm. Find its circumference and its area. Give both answers in terms of π.

  1. 1

    radius = 10 ÷ 2 = 5 cm [1]

  2. 2

    C = π d = π × 10 [1]

  3. 3

    C = 10π cm [1]

  4. 4

    A = π r² = π × 5² = 25π cm² [1]

2. Working back to the radius

Sometimes the circumference or the area is given and you need the radius. Put the given value equal to the formula, then rearrange.

From the area, divide by π and then take the square root. Do not forget the square root: r² = 36 gives r = 6.

r = C/2π

r = √(A/π)

Examiner insight

“Otherwise 25, rather than √25 was often taken as the radius.”

Worked example 12 marks

The area of a circle is 36π cm². Find the radius of the circle.

  1. 1

    π r² = 36π, so r² = 36 [1]

  2. 2

    r = 6 cm [1]

3. Arc length and sector area

A sector is a slice of a circle. Its arc is the part of the circumference on its edge. The sector is a fraction of the whole circle. The fraction is the sector angle over 360°. Find the fraction, then multiply it by the circumference for the arc length, or by the area for the sector area.

Do not mix the two. Arc length is a length, so it uses the circumference formula. Sector area uses the area formula.

arc length = θ/360 × 2π r

sector area = θ/360 × π r²

Examiner insight

“Use of the formula for the area of the sector was occasionally seen instead of the arc length.” “Other common errors included calculating the arc length instead of sector area.”

Worked example 12 marks

A sector of a circle has sector angle 90° and radius 12 cm. Calculate the arc length of the sector. Give your answer in terms of π.

  1. 1

    arc length = 90/360 × 2 × π × 12 [1]

  2. 2

    = 1/4 × 24π = 6π cm [1]

4. Perimeter of a semicircle or a sector

A semicircle is a sector with angle 180°. It is half a circle.

The perimeter is the whole distance round the shape. It is the curved arc plus the straight edges.

A semicircle has one straight edge, the diameter. A sector has two straight edges, both radii.

P_semicircle = (π d)/2 + d

P_sector = θ/360 × 2π r + 2r

Examiner insight

“Some candidates calculated the length of the major arc but did not add on the two radii for the perimeter.”

Worked example 13 marks

A semicircle has a diameter of 15.8 cm. Calculate the perimeter of the semicircle.

  1. 1

    curved edge = 1/2 × π × 15.8 [1]

  2. 2

    perimeter = 24.81… + 15.8 [1]

  3. 3

    = 40.6 cm [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 two sectors made by the same two radii fill the circle. So the major sector angle is 360° minus the minor sector angle.

Then use the same formulas as before, with the major angle.

θ_major = 360° - θ_minor

Examiner insight

“Some mistakenly thought that they could calculate the required perimeter by subtracting the perimeter of the minor sector from the circumference.”

Worked example 13 marks

A minor sector of a circle of radius 6 cm has sector angle 50°. Find the area of the major sector. Give your answer in terms of π in its simplest form.

  1. 1

    major sector angle = 360 - 50 = 310° [1]

  2. 2

    area = 310/360 × π × 6² [1]

  3. 3

    = 31π cm² [1]

Recap

  • Circumference = 2π r = π d. Area = π r². Use the radius, not the diameter, in the area formula.
  • In terms of π: leave π in the answer. Do not convert it to a decimal.
  • A sector is the fraction θ/360 of the circle.
  • Arc length = θ/360 × 2π r. Sector area = θ/360 × π r².
  • Perimeter of a sector = arc + two radii. Perimeter of a semicircle = half the circumference + the diameter.
  • Extended: the major sector has angle 360° minus the minor sector angle.

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 radius 7 cm. Find its circumference. Give your answer in terms of π.2 marks
  2. A circular table top has diameter 1.2 m. Calculate its area. Give your answer in m² correct to 3 significant figures.3 marks
  3. Do not use a calculator. The circumference of a circle is 30π cm. Find the area of the circle. Give your answer in terms of π.3 marks
  4. A wheel has radius 35 cm. It makes 200 complete turns along a road. Calculate the distance it travels. Give your answer in metres correct to 3 significant figures.4 marks
  5. Do not use a calculator. A sector of a circle has radius 8 cm and sector angle 45°. Find the area of the sector. Give your answer in terms of π.2 marks
  6. Do not use a calculator. A sector of a circle has radius 9 cm and sector angle 120°. Find the arc length of the sector. Give your answer in terms of π.2 marks
  7. Do not use a calculator. A minor sector of a circle has radius 12 cm and sector angle 60°. Find the area of the major sector. Give your answer in terms of π.3 marks
  8. A minor sector of a circle has radius 5 cm and sector angle 72°. Calculate the perimeter of the major sector. Give your answer correct to 3 significant figures.4 marks
  9. Do not use a calculator. A slice is cut from a circular cake of radius 9 cm. The top of the slice is a minor sector. The top of the cake that is left is a major sector with perimeter (18 + 15π) cm. Find the sector angle of the slice. Then find the area of the top of the slice. Give the area in terms of π.5 marks
  10. A garden is a major sector of a circle. The minor sector that completes the circle has sector angle 80°. A fence goes all the way round the edge of the garden. The fence is 100 m long. Calculate the area of the garden. Give your answer correct to 3 significant figures.6 marks

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