IB Maths AI SL Topic 3 — Geometry Toolkit Paper 1 & 2 Circles ~6 min read

Arcs & Sectors

An arc is part of a circle’s circumference; a sector is the “pizza slice” enclosed by two radii and an arc. Both are simply fractions of a whole circle, where the fraction is θ/360. Two formulae cover everything — and both are in the formula booklet.

📘 What you need to know

Arc length: l = (θ/360) × 2πr. A fraction of the circumference.
Sector area: A = (θ/360) × πr². A fraction of the full circle area.
Minor = angle < 180° (the smaller slice). Major = angle > 180° (the larger slice).
Major from minor: major angle = 360° − minor angle. Or take the whole circle and subtract the minor.
Perimeter of a sector = arc length + 2r (the two straight radii).
Use degrees: AI SL works in degrees throughout. Make sure your GDC is in DEG mode for the trig parts.

The two formulae in one picture

Same angle θ on the same circle: left highlights the arc (a curve), right highlights the sector (the enclosed pizza slice). Both are the fraction θ/360 of the whole circle.

In the formula booklet arc length: l = θ360 × 2πr
sector area: A = θ360 × πr²

🧭 Recipe — any arc / sector question

Sketch the slice. Mark the angle θ and the radius r.
Decide what’s asked: arc (curve only), sector area (region), or sector perimeter (arc + 2r).
Pick the right formula and substitute. Keep π exact while working; convert to a decimal at the end if asked.
For “major”: use angle 360° − minor, OR compute the full circle then subtract the minor.
Working backwards (given l or A, find θ or r): substitute knowns, rearrange, solve.

Perimeter of a sector: don’t forget the two straight sides. Perimeter = l + 2r, NOT just the arc length.

Worked examples

WE 1

Arc length — playground roundabout

A child stands on a roundabout of radius 1.5 m. The roundabout rotates through an angle of 80°. How far does the child travel?

Identify: arc length, r = 1.5, θ = 80° l = (θ/360) × 2πr Substitute l = (80/360) × 2π(1.5) = (2/9) × 3π = 6π/9 = 2π/3 ≈ 2.09 m (3 sf) l = 2π/3 m ≈ 2.09 m always simplify the fraction θ/360 first: 80/360 = 2/9. Keeps numbers small and reduces calculator mistakes.

WE 2

Sector area — windscreen wiper

A car windscreen wiper of length 45 cm sweeps through an angle of 120°. Find the area it wipes.

Identify: sector area, r = 45, θ = 120° A = (θ/360) × πr² Substitute A = (120/360) × π × 45² = (1/3) × π × 2025 = 675π ≈ 2120.58 cm² A = 675π cm² ≈ 2120 cm² (3 sf) the wiper is the radius of the sector. Simplify 120/360 = 1/3 first — multiplication by 1/3 of πr² is cleaner than dragging fractions through.

WE 3

Perimeter of a sector — sprinkler area

A garden sprinkler waters a sector-shaped patch of radius 8 m and central angle 60°. Find the perimeter of the watered patch.

Perimeter = arc length + 2 × radius Step 1 — arc length l = (60/360) × 2π(8) = (1/6) × 16π = 8π/3 ≈ 8.378 m Step 2 — add the two radii P = 8π/3 + 2(8) = 8π/3 + 16 ≈ 24.4 m (3 sf) P = 8π/3 + 16 ≈ 24.4 m a common trap: students give just the arc length (8.38 m) and forget the two straight radii. The patch has THREE edges — one curved, two straight.

WE 4

Working backwards — find the angle

A wedge of cheese, viewed from above, is a sector of radius 9 cm with a curved edge of length 6π cm. Find the central angle of the wedge.

Use the arc-length formula, solve for θ l = (θ/360) × 2πr Substitute l = 6π, r = 9 6π = (θ/360) × 2π(9) 6π = (θ/360) × 18π Cancel π on both sides 6 = 18θ/360 θ = (6 × 360)/18 θ = 2160/18 = 120° θ = 120° when the arc is given as a multiple of π, the π cancels nicely and you’re left with a clean numerical equation. Always cancel π before multiplying out 360.

WE 5

Major sector — remaining flowerbed

A circular flowerbed has radius 5 m. A minor sector of angle 70° is fenced off for new planting; the rest is left for the lawn. Find the area of the lawn portion (the major sector).

Method 1 — use major angle major angle = 360 − 70 = 290° A = (290/360) × π × 5² = (29/36) × 25π = 725π/36 ≈ 63.3 m² (3 sf) Method 2 — total area minus minor total = π(5²) = 25π minor = (70/360)(25π) = 175π/36 major = 25π − 175π/36 = 725π/36 ✓ major sector area = 725π/36 ≈ 63.3 m² both methods give the same answer. Use whichever feels cleaner — usually “total minus minor” is faster on a calculator since 25π and the minor are quick to compute.

WE 6

Find the radius from a given area

A sector has area 24π cm² and central angle 60°. Find the radius of the sector.

Use sector area formula, solve for r A = (θ/360) × πr² Substitute A = 24π, θ = 60 24π = (60/360) × π × r² 24π = (1/6) × π × r² Divide both sides by π, then ×6 24 = r²/6 r² = 144 r = 12 cm (reject negative) r = 12 cm a radius is a length, so reject the negative root. Always check: (60/360) × π × 144 = (1/6) × 144π = 24π ✓

💡 Top tips

Simplify θ/360 first: 80/360 = 2/9, 60/360 = 1/6, 90/360 = 1/4. Smaller numbers = fewer slips.
Keep π exact while working; convert to a decimal at the end only if the question wants it.
Perimeter ≠ arc length. Perimeter of a sector = arc + 2r.
For “major” questions, either use angle 360 − minor, or compute total then subtract the minor area. Either works.
The formula booklet has both formulae in the geometry & trigonometry section — no need to memorise, but learn to apply them fast.

⚠ Common mistakes

Mixing the two formulae: arc uses 2πr (circumference); area uses πr² (full area). One is a length, the other is an area.
Forgetting the two radii when asked for the perimeter of a sector. Arc alone is not the perimeter.
Using r where r² is needed (or vice versa). Squaring 8 gives 64, not 16.
Using major angle for a minor sector: always check whether the angle given is the one inside the sector you want.
Calculator in RADIAN mode: AI SL works in degrees. Make sure DEG is set before any trig substitution.

That’s the end of the Geometry Toolkit. You can now find midpoints and distances on the plane, build perpendicular bisectors, and handle any arc / sector calculation. Next up in geometry will be Voronoi diagrams — which use these perpendicular bisectors to divide a region into “nearest-neighbour” territories.

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