IB Maths AA SL Topic 3 — Geometry & Trig Paper 1 & 2 🎯 Skill ~3 min practice

AA SL Arc Length & Sector Area skills

A circle slice has two measurements: the curved edge (arc length) and the pie-shape area (sector area). Two formulas, one rule — angles must be in radians. Get that right and the rest is plug-and-play.

The Method

Formula 1

Arc length

l = r θ

curved edge of the slice · units match r

Formula 2

Sector area

A = 12 r² θ

pie-shape area · units squared

Check the angle is in radians. If it’s in degrees, convert it first using θ × π/180.
Pick the right formula — arc length needs l = rθ, area needs A = ½r²θ.
Substitute and solve. Keep π in for exact answers, or use the GDC for decimal answers.

The sector setup

A sector is bounded by two radii (length r) and an arc (length l). The angle θ at the centre — in radians — controls the size of both.

⚠️

Both formulas need radians

If the angle is given in degrees, convert it first: θ (rad) = θ (deg) × π/180. Forgetting this is the single biggest cause of wrong answers in this topic.

Worked examples

WE 1 EASY

A sector has radius r = 8 cm and angle θ = π3 radians. Find the arc length.

step 1 — angle already in radians ✓step 2 — apply l = r θ l = 8 × π/3step 3 — simplify l = 8π/3arc length = 8π/3 cm leave π in for exact form — never decimal unless asked!

WE 2 MEDIUM

A sector has radius 10 cm and angle 1.2 radians. Find the sector area, to 3 sf.

step 1 — angle in radians ✓step 2 — apply A = ½ r² θ A = ½ × 10² × 1.2 = ½ × 100 × 1.2 = 50 × 1.2A = 60 cm² (exact) no π in the angle → clean numerical answer!

WE 3 HARD

A sector has radius 6 cm and angle 75°. Find the arc length and sector area, in exact form.

step 1 — convert to radians 75° × π/180 = 75π/180 = 5π/12part (a) — arc length l = r θ = 6 × 5π/12 = 30π/12 = 5π/2 l = 5π/2 cmpart (b) — sector area A = ½ × 6² × 5π/12 = ½ × 36 × 5π/12 = 18 × 5π/12 = 90π/12 A = 15π/2 cm² always convert degrees first, then use the formulas — never plug degrees in directly!

Practice questions

Try each one yourself first, then click the question to reveal the worked answer. Always check radians vs degrees before substituting.

Q1 EASY A sector has r = 5, θ = π4. Find the arc length. Show answer ▼Hide answer ▲

l = 5 × π/4 l = 5π/4

Q2 EASY A sector has r = 4, θ = 2 radians. Find the sector area. Show answer ▼Hide answer ▲

A = ½ × 4² × 2 = ½ × 16 × 2 A = 16 (units²)

Q3 MEDIUM A sector has radius 12 cm and arc length 8 cm. Find the angle θ. Show answer ▼Hide answer ▲

rearrange l = r θ → θ = l / r θ = 8 / 12 = 2/3 θ = 2/3 radians no units on radians — they’re a pure ratio!

Q4 MEDIUM A sector has radius 9 cm and angle 60°. Find the sector area in exact form. Show answer ▼Hide answer ▲

convert 60° to radians 60 × π/180 = π/3 A = ½ r² θ A = ½ × 81 × π/3 = 81π/6 = 27π/2 A = 27π/2 cm²

Q5 HARD A sector has area 20 cm² and angle π5 radians. Find the radius (3 sf). Show answer ▼Hide answer ▲

A = ½ r² θ → solve for r 20 = ½ × r² × π/5 20 = r² π/10 r² = 200/π r = √(200/π) ≈ 7.978 r ≈ 7.98 cm (3 sf) working backwards — rearrange the formula first, substitute last!

⚠ Common mistakes

Plugging degrees straight into the formula. Both formulas need radians. 90° in l = rθ gives 90r — completely wrong.
Mixing the two formulas. Arc length is r times θ, area is r² times θ (and there’s a ½). The square is on the radius, not theta.
Forgetting the ½ in the area formula. A = ½r²θ — drop the ½ and your answer is doubled.
Wrong units. Arc length matches the units of r (cm). Sector area squares them (cm²). Don’t write cm³ for area.
Decimal when exact form is asked. If θ is a multiple of π, leave π in the answer. 8π/3 ≠ 8.378.
Treating “perimeter of sector” as just the arc. Perimeter = arc + 2 radii (the two straight edges). Don’t forget the bounded edges.

📖

Want the theory?

Read the full Radian Measure notes for why arc length is rθ, the link to the unit circle, and the area-of-segment trick (sector minus triangle).

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