IB Maths: Radians, Arcs & Sectors FAQ

Most lost marks come from calculator mode, not concept. Students leave their GDC in degree mode, apply s = rθ or A = ½r²θ, and get an answer wrong by a factor of roughly 57 — the ratio between radians and degrees. The formulas only work when θ is in radians.

A radian is the angle subtended at a circle's centre by an arc equal in length to the radius, so 2π radians make a full turn. IB Maths uses radians because s = rθ and A = ½r²θ — the arc length and sector area formulas — only hold true when θ is measured in radians.

Convert using π radians = 180°. Multiply degrees by π/180 to get radians, or radians by 180/π to get degrees. So 60° becomes π/3 radians, and 2 radians becomes about 114.6°. Most GDCs have a built-in degree/radian toggle — check yours before every trigonometry question.

Arc length is s = rθ, where r is the radius and θ is the angle in radians. For a circle of radius 8 cm subtending 1.2 radians at the centre, the arc length is s = 8 × 1.2 = 9.6 cm. No degree version is needed once you're working in radians.

Sector area is A = ½r²θ, with θ again in radians. Using the same circle — radius 8 cm, angle 1.2 radians — the sector area is A = ½ × 8² × 1.2 = 38.4 cm². Mixing this up with the arc length formula is one of the most common slips on Paper 1.

A segment is the region between a chord and the arc, so its area equals the sector area minus the triangle area: Area = ½r²(θ − sin θ). For r = 8 cm and θ = 1.2 radians, that's ½ × 64 × (1.2 − sin 1.2) ≈ 8.58 cm². It appears mainly on HL papers and AA SL Paper 2.

Identify r and θ from the diagram first, converting θ to radians if it's given in degrees. Then apply s = rθ for arc length and A = ½r²θ for area — both use the same θ, so work it out once and reuse it. Check your GDC is in radian mode before you start.

Leaving the calculator in degree mode is the single biggest radian error in mock papers. It doesn't just affect trig ratios — it silently breaks s = rθ and A = ½r²θ, producing plausible-looking but wrong answers, off by a factor of about 57.3. Check your mode setting before every geometry question.

It depends on how the angle is given. If a question states an angle without a degree symbol, work in radians throughout; if it uses °, give your final answer in degrees. For s = rθ and sector area, θ must be in radians regardless — convert first, then answer in whatever unit was asked for.

The core content — radian measure, s = rθ, A = ½r²θ — is identical for SL and HL in both Analysis & Approaches and Applications & Interpretation. What changes is how it's combined with other topics: HL questions more often fold radians into trig identities or calculus, pushing mark allocations from around 4 up to 8 per question.

Yes — radian measure, arc length and sector area sit in Topic 3: Geometry and Trigonometry, compulsory for both Analysis & Approaches and Applications & Interpretation, at SL and HL. According to the IB Mathematics guide, this content was first examined in 2021 and remains current for 2025 exams, so it isn't going anywhere.

The fastest fix is targeted past-paper practice on Topic 3 questions, not re-reading notes. Have your child work through 8–10 mixed arc and sector problems under timed conditions, checking GDC mode before each one, then mark against the official mark scheme to see exactly where marks were lost.

You'll find topic-specific Revision Notes covering radian measure, arc length and sector area, a Topical Worksheet with graded practice questions, and full Mock Papers for timed practice. Together they take your child from learning the formulas to applying them under real exam conditions — exactly the practice most students skip until it's too late.