This small topic focuses on two key ideas in circle geometry:
length of an arc and area of a sector.
At SL you will work entirely in degrees (radians are not required).
| Quantity | Formula in degrees |
|---|---|
| Arc length | For a circle of radius r and central angle θ (in degrees): arc length = (θ ÷ 360) × 2πr. |
| Area of a sector | For the same circle and angle: sector area = (θ ÷ 360) × πr2. |
📌 1. Understanding arcs and sectors
- Arc
A piece of the circumference of a circle. If you think of the circle as a track, an arc is a portion of that track. - Sector
The region bounded by two radii and the arc between them. It looks like a “pizza slice”. - The full circle has central angle 360°. If a sector has angle θ, it represents the fraction θ ÷ 360 of the whole circle.
- Both formulas above come from this idea of a fraction:
- Full circumference = 2πr, so arc length is that multiplied by θ ÷ 360.
- Full area = πr2, so sector area is that multiplied by θ ÷ 360.
📌 2. Short worked example
Example
A circle has radius 7 cm. A sector has central angle 60°. Find:
- Arc length
Arc = (θ ÷ 360) × 2πr
= (60 ÷ 360) × 2π × 7
= (1 ÷ 6) × 14π
= (14π ÷ 6) = (7π ÷ 3) cm
≈ 7.33 cm. - Sector area
Area = (θ ÷ 360) × πr2
= (60 ÷ 360) × π × 72
= (1 ÷ 6) × 49π
= 49π ÷ 6 cm2
≈ 25.66 cm2.
📌 3. Strategy and common mistakes
- Always check the angle unit. At SL you will use degrees, so the fraction is θ ÷ 360, not θ ÷ 2π.
- Confirm that the angle given is actually the central angle at the centre of the circle. Angles at the circumference correspond to double angles at the centre in some contexts, so the question will be clear about which angle you should use.
- Keep π symbol until the final step if the question allows an exact answer. Approximating π too early can cause rounding differences.
- Pay attention to units: lengths in centimetres or metres; areas in square units like cm2 or m2.
🧮 GDC Use
- Store r, θ and π values in your calculator memory when using them repeatedly in one question.
- Use the π key rather than 3.14 to keep results accurate and consistent.
- For multi-step questions (for example when the arc length is then used to work out something else), keep full precision on the GDC and round only in your final written answer.
🌍 Real-World Connections
- Designing curved paths or tracks, where the arc length gives the distance travelled along a bend.
- Calculating the area of circular sectors in engineering, for example the cross-section of partially opened valves or gates.
- Determining the size of slices in circular charts or pie charts, where the central angle represents a proportion of the whole.
📐 IA Spotlight
- Investigate how changing the angle of a sector affects arc length and area while keeping radius fixed, and discuss which changes are linear and which are not.
- Model real circular objects (for example, pizzas, wheels, circular gardens) and compare theoretical sector areas with measurements or photographs.
🔍 TOK Perspective
- The idea that a sector with angle θ represents θ ÷ 360 of a full circle relies on the convention that a full turn is 360°. Why 360 and not another number?
- Our experience of circular motion (wheels, rotations, cycles) influences how we define and use angles. To what extent is this mathematical or cultural?