G18: Arcs & Sectors
Calculate arc lengths, angles and areas of sectors of circles
📋 Key Concepts
Arc: A portion of the circumference of a circle.
Sector: A region bounded by two radii and an arc (like a pizza slice).
Key idea: The fraction of the circle is determined by the angle at the centre.
📝 Fraction of a Circle
Fraction = θ/360
Where θ is the angle at the centre in degrees.
Example 1
What fraction of a circle is a sector with angle 72°?
Solution:
Fraction = 72/360 = 1/5
📝 Arc Length
Formula: The arc length is a fraction of the circumference.
Arc length = (θ/360) × 2πr
Example 2
Find the arc length for radius 15 cm and angle 80°.
Solution:
Arc length = (80/360) × 2π × 15
= (2/9) × 30π
= 60π/9 = 20π/3 cm
Example 3
A sector has radius 7 cm and arc length 11 cm. Find the angle at the centre.
Solution:
11 = (θ/360) × 2π × 7
θ = (11 × 360)/(2π × 7)
θ = 3960/(14π)
θ ≈ 90.1°
📝 Area of a Sector
Formula: The sector area is a fraction of the circle area.
Area of sector = (θ/360) × πr²
Example 4
Find the area of a sector with radius 12 cm and angle 150°.
Solution:
Area = (150/360) × π × 12²
= (5/12) × 144π
= 60π cm²
Example 5
A sector has area 20π cm² and radius 8 cm. Find the angle.
Solution:
20π = (θ/360) × π × 64
θ = (20π × 360)/(64π)
θ = 7200/64
θ = 112.5°
📝 Finding the Radius
Example 6
A sector has angle 60° and arc length 8π cm. Find the radius.
Solution:
8π = (60/360) × 2πr
8π = (1/6) × 2πr
8π = (πr/3)
24π = πr
r = 24 cm
Example 7
A sector has area 27π cm² and angle 120°. Find the radius.
Solution:
27π = (120/360) × πr²
27π = (1/3)πr²
81 = r²
r = 9 cm
📝 Perimeter of a Sector
Perimeter = arc length + 2 × radius (the two straight sides)
Example 8
Find the perimeter of a sector with radius 10 cm and angle 90°.
Solution:
Arc length = (90/360) × 2π × 10 = 5π cm
Perimeter = 5π + 2(10) = 5π + 20 ≈ 35.7 cm
🧮 Worked Examples — Exam Style
Arcs and sectors are fractions of a circle. Every question reduces to one idea: the fraction is angle ÷ 360.
Find the area of a sector with radius 10 cm and angle 72°.
Fraction of circle: 72 ÷ 360 = 1/5
Full circle area: π × 10² = 100π
Sector: 1/5 × 100π = 20π cm² (≈ 62.8 cm²)
M1 for 72/360, M1 for π × 10², A1 for 20π. Leaving π in the answer is usually required unless the question says otherwise.
An arc has length 5π cm in a circle of radius 6 cm. Find the sector angle.
Arc = (θ/360) × 2πr → 5π = (θ/360) × 12π
Divide both sides by π: 5 = 12θ/360 → θ = 5 × 30 = 150°
M1 for setting up the arc formula, A1 for 150°. Reverse questions are just solving the same formula.
Exam wording to expect: "Give your answer in terms of π" or "to 1 decimal place" — read which one is asked.
❓ Practice Questions
Q1: Find the arc length for radius 8 cm and angle 135°.
Q2: Find the area of a sector with radius 14 cm and angle 45°.
Q3: A sector has radius 6 cm and arc length 4π cm. Find the angle.
Q4: A sector has area 15π cm² and angle 60°. Find the radius.
Q5: Find the perimeter of a sector with radius 5 cm and angle 120°.
Q6: Find the perimeter of a semicircle with diameter 14 cm
Q7: A sector of angle 40° has area 8π cm². Find its radius.
✅ Answers
- (135/360) × 2π × 8 = 6π cm
- (45/360) × π × 196 = 24.5π cm²
- 4π = (θ/360) × 2π × 6, θ = 120°
- 15π = (60/360) × πr², r² = 90, r = 3√10 cm
- Arc = (120/360) × 2π × 5 = 10π/3 cm, Perimeter = 10π/3 + 10 cm
🧮 Worked Examples — Harder Exam Style
A park is a sector of a circle, radius 50 m, angle 54°. Fencing costs £12 per metre and covers only the curved edge. Find the cost, to the nearest pound.
Arc = (54/360) × 2π × 50 = 0.15 × 100π = 15π m
Length = 15π ≈ 47.12 m
Cost = 47.12 × 12 = £565.49 → £565
M1 for the arc calculation, M1 for the multiplication, A1 for £565. Applied sector questions always need the rounding stated in the question — 'nearest pound' here.
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. Fraction 135/360 = 3/8. Arc = 3/8 × 2π × 8 = 3/8 × 16π = 6π cm
Q2. Fraction 45/360 = 1/8. Area = 1/8 × π × 14² = 196π/8 = 24.5π cm²
Q3. 4π = (θ/360) × 12π → 4 = 12θ/360 → θ = 4 × 30 = 120°
Q4. 15π = (60/360) × πr² = πr²/6 → r² = 90 → r = √90 = 3√10 cm (≈ 9.49 cm)
Q5. Arc = (120/360) × 2π × 5 = 10π/3. Perimeter = arc + 2 radii = 10π/3 + 10 cm. The two straight radii are the part students forget.
Q6. Semicircle perimeter = arc + diameter = (π × 14 ÷ 2) + 14 = 7π + 14 cm (≈ 36.0 cm)
Q7. 8π = (40/360) × πr² = πr²/9 → r² = 72 → r = 6√2 cm (≈ 8.49 cm)
📌 Key Facts to Memorise
- Arc length = (θ/360) × 2πr · Sector area = (θ/360) × πr²
- Both are "fraction × whole circle" — find the fraction first
- Sector perimeter = arc + 2 radii
- Semicircle perimeter = πr + 2r (arc plus diameter)
- π ≈ 3.14159; calculator papers expect the π button, not 3.14
🗺️ In the Exam
Arcs and sectors: 3–4 marks on Papers 2/3 (calculator). "Reverse" questions (find the angle or radius) earn the extra method marks. Answers in terms of π are exact — a decimal where π was required scores 0 for accuracy.
Links to: circles (G17), area and volume (G16), fractions of amounts (R3).
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Simplify the fraction before calculating
- Check if the answer should be in terms of π or as a decimal
- Perimeter includes the arc plus two radii
- When finding angle or radius, rearrange the formula
- Draw a diagram to visualise the sector
- Check units are consistent throughout
🧠 Problem-Solving Strategies
Problem-Solving
Arc length = (angle/360) x 2 x pi x r. Sector area = (angle/360) x pi x r squared. Segment area = sector area minus triangle area. The sector perimeter includes two radii plus the arc length. Always check if the question asks for arc, sector, or segment.
Multi-Step Problem
A sector has radius 14 cm and angle 120 degrees. Find the arc length, the sector area, and the perimeter of the sector.
Solution: Arc = (120/360) x 2 x pi x 14 = (1/3) x 28pi = 29.3 cm. Area = (120/360) x pi x 196 = (1/3) x 196pi = 205.3 cm squared. Perimeter = 14 + 14 + 29.3 = 57.3 cm.
⚠️ Common Errors
Watch Out!
1. Wrong: Calculating arc length as (angle/360) x pi x r Correct: Arc length = (angle/360) x 2 x pi x r. The 2 is essential since the full circumference is 2 pi r.
2. Wrong: Finding the sector perimeter as just the arc length Correct: The sector perimeter includes the two radii AND the arc. Perimeter = 2r + arc length.
3. Wrong: Using the chord length instead of the arc length in sector calculations Correct: Arc length is the curved part of the circumference. The chord is the straight line joining the arc endpoints.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A sector has arc length 15 cm and radius 10 cm. (a) Find the angle of the sector. (b) Find the sector area. (c) Find the segment area between the chord and the arc.
(a) Arc = (angle/360) x 2 x pi x 10 = 15. angle = 15 x 360/(20pi) = 85.9 degrees.
(b) Sector area = (85.9/360) x pi x 100 = 75.0 cm squared.
(c) Triangle area = 1/2 x 100 x sin(85.9) = 49.9 cm squared. Segment = 75.0 - 49.9 = 25.1 cm squared.
Mark scheme: M1 arc formula rearranged, A1 angle 85.9, M1 sector area, A1 75.0, M1 triangle area, A1 segment 25.1
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A farmer has a sector-shaped field with radius 80 m and angle 60 degrees.
(a) Find the area in hectares (1 hectare = 10,000 m squared).
(b) How much fencing is needed for the curved boundary only?
(c) A neighbour has the same radius but angle 120 degrees. A student says the field is twice as big. Is this correct?
Answers: (a) Area = (60/360) x pi x 6400 = 3351 m squared = 0.335 hectares. (b) Arc = (60/360) x 2 x pi x 80 = 83.8 m. (c) Yes — area is proportional to the angle when the radius is fixed. Doubling the angle doubles the area: 6702 m squared.
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