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GCSE Maths / Edexcel

Arcs and sectors

Name the main parts of a circle and calculate arc lengths and sector areas using the angle fraction.

Geometry and MeasuresFoundation and HigherGrades 5 to 8Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Arcs and sectors

Pearson Edexcel GCSE Maths geometry G9 and G16: identify circle parts and calculate arc length and sector area.

Practise this topicUpload your answerAsk AI to explain thisTry exam-style questions
Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

A radius goes from the centre of a circle to the circumference. A diameter goes all the way across the circle through the centre, so diameter = 2 x radius.

A chord joins two points on the circumference. A diameter is a special chord that passes through the centre.

A tangent is a straight line that touches the circle at exactly one point. This becomes important again in Circle Theorems.

An arc is part of the circumference. A sector is a slice of the circle made by two radii and an arc.

Arc length and sector area both use the same fraction: angle / 360.

Arc length = angle / 360 x circumference = angle / 360 x 2πr.

Sector area = angle / 360 x area of circle = angle / 360 x πr².

A common Edexcel trap is using the diameter as the radius. If the question gives a diameter, halve it before using r.

Key ruleUse the sector fraction angle / 360. Arc length uses 2πr; sector area uses πr².

Diagram guide

diameterradiuschordtangentsectorarc is part of the circumference
Circle vocabularyKnow radius, diameter, chord, tangent, arc and sector before starting the calculations.
thetaarcsectorfraction of circle = angle / 360arc length and sector area use the same fraction
Arc and sector fractionBoth arc length and sector area use the same fraction of the full circle.

Worked examples

Arc length

A sector has radius 8 cm and angle 90 degrees. Find the arc length in terms of π.

thetaarcsectorfraction of circle = angle / 360arc length and sector area use the same fraction
Example: quarter circle arc90 degrees is one quarter of a full turn.
  1. Fraction of circle = 90 / 360 = 1/4.
  2. Full circumference = 2πr = 2π x 8 = 16π.
  3. Arc length = 1/4 x 16π = 4π cm.

Answer: 4π cm

Sector area

A sector has radius 6 cm and angle 60 degrees. Find its area in terms of π.

thetaarcsectorfraction of circle = angle / 360arc length and sector area use the same fraction
Example: sector area60 degrees is one sixth of a full circle.
  1. Fraction of circle = 60 / 360 = 1/6.
  2. Full circle area = πr² = π x 6² = 36π.
  3. Sector area = 1/6 x 36π = 6π cm².

Answer: 6π cm²

Diameter given

A sector has diameter 20 cm and angle 72 degrees. Find the sector area in terms of π.

diameterradiuschordtangentsectorarc is part of the circumference
Example: diameter to radiusHalve the diameter before using the area formula.
  1. Radius = 20 / 2 = 10 cm.
  2. Fraction = 72 / 360 = 1/5.
  3. Full circle area = 100π.
  4. Sector area = 1/5 x 100π = 20π cm².

Answer: 20π cm²

Common mistakes

  • Using angle / 180 instead of angle / 360.
  • Using area formula for arc length or circumference formula for sector area.
  • Forgetting to halve the diameter to get the radius.
  • Rounding too early in calculator questions.
  • Calling a chord a tangent or confusing sector and segment.

Quick exercise

Try these before moving to the exam-style questions.

  1. What fraction of a full circle is 90 degrees?
    thetaarcsectorfraction of circle = angle / 360arc length and sector area use the same fraction
    Quick check: angle fractionCompare the sector angle with 360 degrees.
  2. A sector has radius 5 cm and angle 180 degrees. Find the arc length in terms of π.
    thetaarcsectorfraction of circle = angle / 360arc length and sector area use the same fraction
    Quick check: semicircle arc180 degrees is half the circumference.
  3. A sector has radius 4 cm and angle 90 degrees. Find the sector area in terms of π.
    thetaarcsectorfraction of circle = angle / 360arc length and sector area use the same fraction
    Quick check: sector areaFind one quarter of the circle area.
  4. What is the name of a line that touches a circle at exactly one point?
    diameterradiuschordtangentsectorarc is part of the circumference
    Quick check: circle vocabularyThis word is vital for Circle Theorems.
  5. A circle has diameter 14 cm. What radius should you use in sector formulae?
    diameterradiuschordtangentsectorarc is part of the circumference
    Quick check: diameterRadius is half the diameter.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleFoundation and HigherCalculator3 marks

A sector has radius 12 cm and angle 90 degrees. Work out the arc length in terms of π.

thetaarcsectorfraction of circle = angle / 360arc length and sector area use the same fraction
Question diagram: arc lengthArc length is a fraction of the circumference.
arc lengthcircle sectorangle fraction
Standard exam styleFoundation and HigherCalculator4 marks

A sector has radius 9 cm and angle 120 degrees. Work out the area of the sector in terms of π.

thetaarcsectorfraction of circle = angle / 360arc length and sector area use the same fraction
Question diagram: sector areaSector area is a fraction of the full circle area.
sector arearadiuscircle measures
ChallengeHigherCalculator5 marks

A sector has diameter 16 cm and angle 135 degrees. Work out the arc length to 3 significant figures.

diameterradiuschordtangentsectorarc is part of the circumference
Question diagram: diameter givenUse radius 8 cm, not 16 cm.
arc lengthdiameterrounding