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Mensuration

Mensuration

Circles, arcs and sectors

2D Minor Segment Paper 1 + 2 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Circumference and area
  4. Arcs and sectors
  5. Working backwards and shaded areas
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 5.3: circles, arcs and sectors
  • 5.3Find the circumference and area of a circle.
  • 5.3Find arc lengths and sector areas as fractions of the circle, including major sectors.
  • 5.3Give answers in terms of \(\pi\), and work backwards to an angle or radius.

Before you start: Units, perimeter and area. Next: Prisms and cylinders.

Formula bank

CircumferenceOn the formula sheet\(C=2\pi r=\pi d\)Use it when distance round a circle.
Area of a circleOn the formula sheet\(A=\pi r^{2}\)
Arc lengthOn the formula sheet\(\dfrac{\theta}{360}\times2\pi r\)θ = the angle at the centre.Use it when part of the circumference.
Sector areaOn the formula sheet\(\dfrac{\theta}{360}\times\pi r^{2}\)Use it when a 'slice' of the circle.

Circumference and area

For a circle of radius 5 cm: circumference \(2\pi\times5=10\pi\approx31.42\) cm; area \(\pi\times5^{2}=25\pi\approx78.54\) cm². 'In terms of \(\pi\)' means leave \(\pi\) in the answer (10π, 25π) - no calculator rounding.

Try it yourself

1
Circumference of a circle of radius 5 cm, in terms of \(\pi\)? Type the number in front of π.
Show answer

Answer 10

2
Area of a circle of radius 7 cm, to 2 d.p. (cm²)?
Show answer

Answer 153.94

Arcs and sectors

Sector with angle 80°80°Oarcr = 9
A sector of radius 9 cm and angle 80°.

A sector with angle \(\theta\) is \(\dfrac{\theta}{360}\) of the whole circle, so its arc is that fraction of the circumference and its area is that fraction of the circle's area. The perimeter of a sector = arc + two radii.

General methodHow to find an arc, sector area or sector perimeter

  1. Write the fraction θ/360 (for a MAJOR sector, θ = 360 − the small angle).
  2. Arc = fraction × 2πr. Area = fraction × πr².
  3. Perimeter of the sector = arc + r + r.
  4. Leave π in if asked 'in terms of π'; otherwise round at the end.

Example A sector has radius 9 cm and angle 80°. Find the arc length, the area and the perimeter.

Fraction \(\tfrac{80}{360}\).

  1. arc \(=\tfrac{80}{360}\times18\pi=4\pi\approx12.57\)
  2. area \(=\tfrac{80}{360}\times81\pi=18\pi\approx56.55\)
  3. perimeter \(=4\pi+18\approx30.57\)

Answer 12.57 cm, 56.55 cm², 30.57 cm

Try it yourself

1
Arc length of a sector, radius 10 cm, angle 72° (2 d.p.)?
Show answer

Answer 12.57

2
Area of that sector (2 d.p.)?
Show answer

Answer 62.83

Working backwards and shaded areas

General methodHow to find a missing angle or radius

  1. Write the formula with the unknown: e.g. (θ/360) × πr² = given area.
  2. Cancel π if it appears on both sides.
  3. Solve for θ (or for r², then square-root).

Example A sector of radius 12 cm has area \(24\pi\) cm². Find its angle.

\(\tfrac{\theta}{360}\times144\pi=24\pi\).

  1. \(\tfrac{\theta}{360}=\tfrac{24}{144}\)
  2. \(\theta=60\)

Answer 60°

Shaded parts: subtract. A segment = sector − triangle; a ring (annulus) = big circle − small circle (\(R=10\), \(r=6\): \(100\pi-36\pi=64\pi\)).

Ways they test it

Type 1 Perimeter of a sector

"Calculate the perimeter of the major sector, radius 6 cm, minor angle 110°." (3 marks)

Spot it 'perimeter' and a sector.

  1. Major angle 250°.
  2. Arc + 2 radii.

For example \(\tfrac{250}{360}\times12\pi\)

  1. 26.18 + 12

Answer 38.18 cm

Type 2 In terms of π

"Find the area of the sector. Give your answer in terms of \(\pi\)." (2 marks)

Spot it 'in terms of π'.

  1. Keep π as a symbol; simplify the fraction.

Type 3 Find the angle

"The arc length is \(5\pi\) cm and the radius 9 cm. Find the angle." (2 marks)

Spot it arc or area given.

  1. Formula = value; cancel π; solve.

For example \(\tfrac{\theta}{360}\times18\pi=5\pi\)

    Answer 100°

    Type 4 Shaded area

    "Find the shaded area" (segment, ring, semicircle in a square). (3-4 marks)

    Spot it a shaded region.

    1. Big area minus small area.

    Mistakes to avoid

    • ✗ Sector perimeter = arc only✓ arc + 2 radiiPerimeter is the whole boundary.
    • ✗ Using the minor angle for a major sector✓ 360 − minor angleMajor = the bigger part.
    • ✗ Area with \(2\pi r\)✓ \(\pi r^{2}\)Circumference is a length, area is squared.

    Summary

    • \(C=2\pi r\), \(A=\pi r^{2}\).
    • Arc = \(\tfrac{\theta}{360}\times2\pi r\); sector = \(\tfrac{\theta}{360}\times\pi r^{2}\).
    • Sector perimeter = arc + 2 radii; major sector uses 360 − minor angle.
    • Shaded areas: big minus small.

    If you see …

    • "in terms of π"keep π in the answer.
    • an arc or area givenformula = value, solve for θ or r.

    Practice

    Practice: easy, medium, hard

    Easy

    1
    Circumference of a circle of radius 5 cm (2 d.p.)?
    Show answer

    Answer 31.42

    2
    Area of a circle of radius 5 cm (2 d.p.)?
    Show answer

    Answer 78.54

    3
    Perimeter of a semicircle of radius 4 cm (2 d.p.)?
    Hint
    Half the circumference + the diameter.
    Show answer

    Answer 20.57

    Medium

    1
    Sector, radius 9 cm, angle 80°. Perimeter (2 d.p.)?
    Guide me

    What is it asking? a perimeter

    What do we know? arc \(4\pi\)

    Plan + 18

    Show working
    1. \(4\pi+18=30.57\)

    Answer 30.57

    2
    Sector of radius 12 cm with area \(24\pi\) cm². Angle?
    Guide me

    What is it asking? an angle

    What do we know? \(\tfrac{\theta}{360}\times144\pi=24\pi\)

    Plan cancel π

    Show working
    1. \(\theta=\tfrac{24}{144}\times360\)

    Answer 60

    3
    Area of a ring between circles of radius 10 cm and 6 cm, in terms of π. Type the number in front of π.
    Guide me

    What is it asking? an area

    What do we know? \(100\pi-36\pi\)

    Plan subtract

    Show working
    1. \(64\pi\)

    Answer 64

    Hard

    1
    Perimeter of the major sector, radius 6 cm, minor angle 110° (2 d.p.)?
    Guide me

    What is it asking? a perimeter

    What do we know? major angle 250°

    Plan arc + 12

    Show working
    1. \(\tfrac{250}{360}\times12\pi=26.18\)
    2. 26.18 + 12 = 38.18

    Answer 38.18

    PartAnswer / workingMarksGuidance
    250°B1
    \(\tfrac{250}{360}\times2\pi\times6\)M1
    + 12M1
    38.18A1
    Total: 4 marks
    2
    An arc has length \(5\pi\) cm and radius 9 cm. Find the angle.
    Guide me

    What is it asking? an angle

    What do we know? \(\tfrac{\theta}{360}\times18\pi=5\pi\)

    Plan solve

    Show working
    1. \(\theta=\tfrac5{18}\times360=100\)

    Answer 100

    PartAnswer / workingMarksGuidance
    equationM1
    100A1
    Total: 2 marks
    3
    A quarter circle of radius 6 cm has the triangle joining its ends and centre removed. Area of the segment left (2 d.p.)?
    Guide me

    What is it asking? an area

    What do we know? sector \(9\pi\)

    Plan minus triangle ½ × 6 × 6

    Show working
    1. 28.27 − 18 = 10.27

    Answer 10.27

    PartAnswer / workingMarksGuidance
    \(9\pi\)M1
    − 18M1
    10.27A1
    Total: 3 marks

    Can I do this?

    Anything unticked: revise that section first. All ticked: take the quiz.

    Quiz

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