A sector is a fraction of the full circle area: A sector with angle θ covers a fraction θ/360 of the full circle area (πr²).
Interactive: switch to ''Sector Area'' tab to see the shaded area change with angle.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Worked example — sector area
Find the area of a sector with radius 10 cm and angle 108°.
Step by step
- Substitute.
- Evaluate.
Final answer
Area = 30π ≈ 94.2 cm².
Free preview
This is the free notes preview
You're reading the free notes. Aimnova Pro unlocks the full study experience — and you can try it with your first topic free to keep:
- FlashcardsLock in vocabulary and key terms with spaced repetition.
- Practice questionsAnswer exam-style questions and get instant AI marking.
- Mock exams & past-paper vaultSit full mocks and see exactly how examiners award marks.
- Personalised study planA daily plan built around your exam date and weak areas.
Worked example — find the angle
A sector of radius 8 cm has area 32π cm².
Find its angle.
Step by step
- Substitute into the area formula.
- Cancel π and rearrange.
- Solve for θ.
Final answer
Angle = 180° (a semicircle).
Practice with real exam questions
Answer exam-style questions and get AI feedback that shows you exactly what examiners want to see in a full-marks response.
Worked example — find both arc and area
A sector has radius 6 cm and angle 150°.
Find:
(a) the arc length,
(b) the area.
Step by step
- (a) Arc length.
- (b) Sector area.
Final answer
Arc length ≈ 15.7 cm; Area ≈ 47.1 cm².
IB-style question — an annular sector (a ring)
A metal washer is shaped like an annular sector: a sector of a large circle with a smaller sector removed. The outer radius is 8 cm, the inner radius is 5 cm, and the central angle is 120°.
Find the area of the annular sector.
Step by step
- Take the big sector and subtract the small sector — both share the 120° angle, so subtract the radii squared.
- Substitute R = 8, r = 5, θ = 120°.
- Evaluate.
Final answer
A = 13π ≈ 40.8 cm².
The annular sector from the question: outer radius R = 8, inner radius r = 5, angle 120°. Its area is the big sector minus the small one.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Worked example — pizza slice
A circular pizza has diameter 30 cm.
It is cut into 8 equal slices.
Find the area of one slice.
Step by step
- Each slice has angle θ = 360/8 = 45°. Radius = 15 cm (half of diameter).
- Area of one slice.
Final answer
Area of one slice ≈ 88.4 cm².
IB-style question — a curved track (area → volume)
A curved section of running track is an annular sector with outer radius 30 m, inner radius 24 m and central angle 100°.
(a) Find the area of the track surface.
(b) The surface is covered with rubber 0.05 m deep. Find the volume of rubber needed.
Step by step
- (a) Annular-sector area = (θ/360) × π(R² − r²) with R = 30, r = 24, θ = 100°.
- Evaluate the area.
- (b) The rubber is a thin layer: volume = surface area × depth.
Final answer
(a) ≈ 283 m². (b) ≈ 14.1 m³.
The track is an annular sector (R = 30 m, r = 24 m, angle 100°). Find its area, then multiply by the 0.05 m depth for the volume of rubber.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.