IB Maths AA HL
Topic 3 — Geometry & Trigonometry
Paper 1 & 2
~6 min read
Arcs & Sectors Using Radians
When the angle is measured in radians, the arc length and sector area formulas collapse to their cleanest possible forms: l = rθ and A = ½r2θ. No factor of 360°, no fraction-of-the-whole gymnastics. Both formulas are given in the formula booklet, and both should be your default whenever the angle is in radians — which, in IB AA HL, is most of the time.
📘 What you need to know
- Arc length: l = rθ, where θ is in radians.
- Sector area: A = ½r2θ, where θ is in radians.
- Both are in the formula booklet in the Geometry & Trigonometry section — you don’t need to memorise them, but you do need to know when to reach for them.
- Radians only: these formulas are wrong if the angle is in degrees. Convert to radians first.
- Perimeter of sector = arc + 2r = rθ + 2r = r(θ + 2).
- Major sector/arc: if θ is the minor angle, the major angle is 2π − θ.
- Reverse problems: any one of θ, r, l, A can be the unknown — just rearrange the formula.
- If two unknowns: combine arc length and area equations to solve simultaneously.
Length of an arc
The arc length formula in radians is as simple as it gets — just multiply the radius by the angle.
Arc length (radians)
l = rθ
This is the very definition of radian measure: 1 radian is the angle whose arc on a unit circle has length 1. Scale up to a circle of radius r and the arc length scales with it — hence l = rθ. No fractions of 360 to worry about.
Area of a sector
Almost as clean — half the radius squared, times the angle.
Sector area (radians)
A = 12 r2θ
Why the ½? When θ = 2π (a full revolution), this should give the area of the full disc — and ½r2(2π) = πr2 ✓. The formula is just the natural radian rewrite of A = (θ/2π) × πr2, with the 2π cancelling cleanly.
Radians vs degrees — why bother switching
Arc length
l = rθ (radians)
vs (θ/360) × 2πr in degrees
Sector area
A = ½r2θ (radians)
vs (θ/360) × πr2 in degrees
Both radian formulas are shorter, faster to substitute into, and easier to manipulate algebraically. They’re also the only forms used elsewhere in the IB course — particularly in calculus, where derivatives of trig functions only behave nicely with radians.
Mental model: in radians, the arc length is the radius multiplied by “how many radii of arc you’ve travelled”. An angle of 1 rad = 1 radius of arc; angle of 2 rad = 2 radii of arc; angle of π rad = π radii of arc (a semicircle). The formula l = rθ just makes this literal.
🧭 Recipe — solving radian arc & sector problems
- Check the angle is in radians. If it’s in degrees, convert: multiply by π/180.
- Sketch the situation. Label the angle, radius, and what’s asked for.
- Pick the right formula: l = rθ for arc length, ½r2θ for sector area, or both for perimeter / composite problems.
- Substitute carefully. Keep θ as an exact fraction of π for “nice” angles; keep decimals to 4+ s.f. during calculation.
- Solve and round. Final answer to 3 s.f. unless told otherwise; leave answers in terms of π for exact-form requests.
- For reverse problems: write the formula, substitute knowns, rearrange. For two unknowns, solve simultaneously using both arc and area equations.
Worked examples
A sector of a circle has central angle 2π/5 radians and radius 12 cm. Find the length of the arc, leaving your answer in exact form.
Step 1: Substitute into l = rθ
l = 12 × 2π/5
Step 2: Simplify
l = 24π/5
Step 3: Decimal value (if needed)
24π/5 ≈ 75.398/5 ≈ 15.08…
l = 24π/5 cm ≈ 15.1 cm (3 s.f.)
no division by 360 — just multiply radius by angle
A sector has central angle 3π/4 radians and radius 8 cm. Find the area of the sector in exact form.
Step 1: Substitute into A = ½r²θ
A = ½ × 8² × 3π/4
Step 2: Simplify
A = ½ × 64 × 3π/4
A = 32 × 3π/4 = 96π/4 = 24π
Step 3: Decimal value
24π ≈ 75.398…
A = 24π cm² ≈ 75.4 cm² (3 s.f.)
3π/4 is three-eighths of a full turn (2π), so the sector is three-eighths of the disc — sanity check: (3/8) × 64π = 24π ✓
WE 3Perimeter of a sector with a decimal angle
A sector has central angle 1.2 radians and radius 5 cm. Find the perimeter of the sector.
Step 1: Find the arc length
l = rθ = 5 × 1.2 = 6 cm
Step 2: Add the two radii
P = arc + 2r = 6 + 2(5)
P = 6 + 10 = 16
P = 16 cm
decimal radians work just as well — the formula doesn’t care whether θ is a multiple of π or not
WE 4Find the angle from a known arc length
An arc of a circle of radius 4 cm has length 9 cm. Find the angle subtended at the centre, in (a) radians, and (b) degrees, correct to 3 s.f.
(a) Use l = rθ and rearrange for θ
θ = l/r = 9/4 = 2.25
(a) θ = 2.25 radians
(b) Convert to degrees: multiply by 180/π
2.25 × 180/π = 405/π
≈ 128.916…
(b) θ ≈ 129° (3 s.f.)
2.25 rad > π/2 (≈1.57) but < π (≈3.14), so the angle is between 90° and 180° — matches our 129°
WE 5Find the radius from a known sector area
A sector with central angle π/3 radians has area 30 cm². Find the radius, correct to 3 s.f.
Step 1: Set up the equation A = ½r²θ
30 = ½ × r² × π/3
30 = πr²/6
Step 2: Solve for r²
r² = 30 × 6/π = 180/π
Step 3: Take positive square root (r > 0)
r = √(180/π) ≈ √(57.296…)
r ≈ 7.5697…
r ≈ 7.57 cm (3 s.f.)
always discard the negative root — radius is non-negative
WE 6Pendulum sweep — arc length and area together
A pendulum of length 50 cm swings through a total angle of 0.4 radians at its widest. Find (a) the distance travelled by the bob from one extreme to the other, and (b) the area swept out by the pendulum on a single swing.
Identify: the bob traces an arc of a circle with r = 50, θ = 0.4
(a) Distance travelled = arc length
l = rθ = 50 × 0.4 = 20
(a) l = 20 cm
(b) Area swept = sector area
A = ½r²θ = ½ × 50² × 0.4
A = ½ × 2500 × 0.4 = 1250 × 0.4
A = 500
(b) A = 500 cm²
both answers come out to clean integers — a hint that the question is well-set for radians, not degrees
💡 Top tips
- Default to the radian formulas. They’re cleaner, in the formula booklet, and match the angle conventions used everywhere else in IB AA HL.
- If the angle is in degrees, convert first. Don’t try to use l = rθ with a degree value — the answer will be wrong by a factor of π/180.
- Keep θ exact when possible: π/3, 2π/5, etc. produce cleaner working than decimal substitutes like 1.047 or 1.257.
- Perimeter shortcut: P = r(θ + 2). Useful when you want to factor out r directly.
- Two-unknown problems: if you don’t know r or θ, look for two equations — usually one from arc length and one from sector area. Solve simultaneously.
- Check whether the answer asks for arc, perimeter, or area. They’re three different quantities and the most common reason for “right method, wrong answer” mistakes.
- Use the formula booklet in the exam — both formulas are listed under Geometry & Trigonometry. No need to memorise, but recognise instantly.
⚠ Common mistakes
- Using the radian formula with a degree value. l = rθ is wrong if θ is in degrees. Convert first.
- Forgetting the ½ in the sector area formula. Without it, you’d be calculating r2θ — twice the actual area.
- Forgetting to square the radius in the area formula.
- Adding 2r to the arc length when only the arc was asked for. Read the question carefully — perimeter and arc length are different.
- Forgetting to take the square root when solving for the radius from a sector area.
- Confusing the formula booklet versions. The booklet gives l = rθ and A = ½r2θ — the radian forms only.
- Using minor angle for major sector (or vice versa). For the major arc/sector, use 2π − θ (not 360° − θ) since you’re working in radians.
And that closes the Geometry Toolkit — the four foundational notes of Topic 3. You’ve now got coordinate geometry’s three big formulas, both arc/sector formulas in degrees and radians, and the radian conversion machinery to switch between them. Everything beyond this in Topic 3 — right-angled and non-right-angled trigonometry, the unit circle, identities, equations, and 3D geometry — leans on this toolkit. Get these solid and the rest of trigonometry becomes a series of applications, not new concepts to wrestle with.
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