Key Idea: Radians are the natural way to measure angles, and they unlock two clean circle formulas — arc length and sector area. These come up on Paper 1 (non-calculator), so the working has to be exact and by hand.
🔄 Radians ↔ degrees
- a half turn (180°); a full turn is 2π rad
- the angle whose arc length equals the radius
An angle written as a multiple of π (or just a plain number like 1.5) is in radians. A full turn is 2π ≈ 6.28, so radian angles are small numbers. On Paper 2, set your GDC to radian mode — sin(30) in radian mode is not sin(30°).
⭕ Arc length & sector area
- arc length
- radius
- central angle — must be in radians
- sector area
- radius
- central angle in radians
✏️ IB-style worked examples
IB-style question — convert between degrees and radians
Convert (a) 75° to radians, and (b) 5π/6 radians to degrees.
Step by step:
(a) Multiply by π/180 and cancel.
(b) Multiply by 180/π — the π cancels.
Final answer:
(a) 5π/12; (b) 150°.
IB-style question — arc length and sector area
A sector has radius 9 cm and central angle π/3 radians. Find (a) the arc length and (b) the area of the sector.
Step by step:
(a) Use s = rθ.
(b) Use A = ½r²θ. Square the radius first.
Simplify.
Final answer:
(a) 3π ≈ 9.42 cm; (b) 27π/2 ≈ 42.4 cm².
Important: These formulas are built for radians only. Plugging a degree angle straight in gives a wildly wrong answer. If the angle is in degrees, convert to radians first (× π/180) — then substitute.
Tap each card to reveal the answer.
Exam Tips
- 180° = π rad: deg → rad multiply by π/180; rad → deg multiply by 180/π.
- Learn the key six: 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 180°=π, 360°=2π.
- Arc s = rθ and sector A = ½r²θ — both with θ in radians (both in the booklet).
- Perimeter of a sector = arc + 2r; segment = sector − triangle (½r²θ − ½r²sin θ).
- Working backwards? Substitute the known values, then solve for r or θ.