Key ideas
- An arc is part of the circumference. A sector is a slice of the circle, like a slice of pizza.
- The angle at the centre tells you what fraction of the circle you have: angle360.
- Arc length = angle360 × πd Sector area = angle360 × πr²
- Arc length uses the diameter (or 2 × radius). Sector area uses the radius, squared.
- The perimeter of a sector is the arc length plus the two radii.
- A major arc or sector is the bigger part. Its angle is 360° take away the smaller angle.
- Exam questions are usually 3 marks: •1 the fraction, like 106360, •2 putting the numbers into the right formula, •3 the answer. Give your answer to 1 decimal place unless told otherwise.
Arc length
The sector has radius 9.15 m and angle 106°. Calculate the length of the arc.(3 marks)
Fraction of the circle.106360✓1
In plain EnglishThe sector is 106° out of the full 360°, so the arc is 106/360 of the whole circumference.
Substitute into the formula.Arc = 106360 × π × d= 106360 × π × 18.3✓2
In plain EnglishThe arc length formula uses the diameter: d = 2 × 9.15 = 18.3 m.
Calculate.= 16.928…Arc = 16.9 m✓3
In plain EnglishType it all in one go on your calculator, then round at the end.
⚠ A common mistake: using π × 9.15 (the radius) instead of π × 18.3 (the diameter) gives 8.5 m. That only gets 2 of the 3 marks.
Sector area
Calculate the area of the sector.(3 marks)
Fraction of the circle.140360✓1
In plain EnglishThe sector is 140° out of 360°.
Substitute into the formula.Area = 140360 × π × r²= 140360 × π × 11²✓2
In plain EnglishArea uses the radius, squared.
Calculate.= 147.829…Area = 147.8 cm²✓3
In plain EnglishSquare units for area.
Perimeter of a sector
Calculate the perimeter of the sector.(3 marks)
Arc length.Arc = 70360 × π × 20✓1= 12.217… cm
In plain EnglishFind the arc first. The radius is 10 cm, so the diameter is 20 cm.
Add the two radii.Perimeter = 12.217… + 10 + 10✓2= 32.2 cm✓3
In plain EnglishThe perimeter is the whole way round the outside: the curved arc and the two straight edges.
⚠ Don't forget the two straight edges. The arc on its own is not the perimeter.
Major arcs and sectors
The angle at the centre of the small sector is 110°. Calculate the length of the major arc (the long way round).(3 marks)
Angle of the major sector.360 − 110 = 250°250360✓1
In plain EnglishThe major arc goes the long way round, so its angle is what is left of 360°.
Substitute.Arc = 250360 × π × 18✓2
In plain EnglishRadius 9 cm, so diameter 18 cm.
Calculate.= 39.3 cm✓3
In plain EnglishType it into the calculator in one go and round to 1 decimal place. It should be more than half the circumference (56.5 cm), which makes sense for the long way round.
Problems in context
A wiper blade sweeps through 140°. It cleans the part of the windscreen between 10 cm and 45 cm from the pivot. Calculate the area cleaned, to the nearest cm².(3 marks)
Area of the big sector.140360 × π × 45² = 2474.00…✓1
In plain EnglishFind the sector the outer end of the blade sweeps out.
Take away the small sector.140360 × π × 10² = 122.17…✓2
In plain EnglishThe part nearest the pivot isn't cleaned, so take its sector away.
Calculate.2474.00 − 122.17 = 2352 cm²✓3
In plain EnglishRound at the very end.
Working back
A sector of radius 8 cm has an arc length of 15 cm. Calculate the angle at the centre.(3 marks)
Write an equation.x360 × π × 16 = 15✓1
In plain EnglishCall the angle x. The arc formula still works: it just has x in it.
Rearrange.x = 15 × 360π × 16✓2
In plain EnglishUndo each step: multiply both sides by 360 and divide both sides by π × 16.
Calculate.x = 107.4°✓3
In plain EnglishRound to 1 decimal place. Check: an arc of 15 cm is just under a third of the circumference (50.3 cm), and 107° is just under a third of 360°.
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Your turn
Calculator allowed. Type the fraction of the circle like 106/360, then your answer to 1 decimal place unless told otherwise. Using π = 3.14 is fine too. Stuck? Tap Need a hint?
Past paper questions
Ready for exam-style questions? Arcs and sectors come up in National 5 most years, usually for 3 marks.
2023 Paper 2 marking instructionsQuestion 3 is an arc length question. The marking shows exactly how the 3 marks are given, including common mistakes.SQA · PDF Open ↗ (opens in a new tab) National 5 arcs and sectors: topic pageEvery National 5 arcs and sectors question, with worked videos.maths.scot Open ↗ (opens in a new tab) Arcs and sectors: past paper questionsPast paper questions on arc length and sector area.Dynamic Maths Open ↗ (opens in a new tab)