IB Maths AA SLTopic 3 — Geometry & TrigPaper 1 & 2~8 min read
Arcs & Sectors Using Degrees
Think of a pizza. The crust on one slice is an arc. The whole slice is a sector. Find either by working out what fraction of the full circle you’ve got — that fraction is just the angle divided by 360°.
📘 What you need to know
An arc is a piece of the circle’s circumference (the curve only).
A sector is a piece of the circle’s area, enclosed by two radii and an arc (the whole pie slice).
Both formulas come from the same idea: take the fraction θ / 360° of the whole.
Arc length: l = θ360 × 2πr · Sector area: A = θ360 × πr2
Neither degree formula is in the formula booklet (only the radian versions are) — so memorise these two.
Arc vs sector — what’s the difference?
This is the first thing students mix up. An arc is just the curve; a sector is the whole filled-in slice. Same idea, different formulas.
📏 ARC = THE CRUST
A length — measured in cm, m, etc.
🍕 SECTOR = THE WHOLE SLICE
An area — measured in cm², m², etc.
A sector of a circle — labelled
The big idea: it’s all just a fraction
A full circle has 360°. If you take a slice with angle θ°, you’re keeping θ / 360 of the full circle. Multiply that fraction by the whole circumference (or whole area) and you’ve got your answer.
🤔 The fraction trick — works every time
Think of it this way. A 90° slice is 90360 = 14 of the circle. So the arc length is 14 of the circumference, and the sector area is 14 of the area. That’s literally what the formula says.
📏
Arc Length
l = θ360 × 2πr
θ = central angle (degrees) r = radius
🍕
Sector Area
A = θ360 × πr2
θ = central angle (degrees) r = radius
Both formulas have the same θ360 at the front. The only difference is what you multiply it by — the circumference for arc length, or the area for sector area.
Minor vs major
If the angle at the centre is less than 180°, you’ve got a “minor” arc/sector — the smaller piece. If the angle is more than 180°, you’ve got a “major” arc/sector — the bigger piece. Same formulas either way, just plug in the actual angle.
Minor (θ < 180°)
The “small slice”
Major (θ > 180°)
The “rest of the circle”
Useful trick: if you only know the minor angle, the major angle is 360° − θ. So a 100° minor sector goes with a 260° major sector — they always add up to 360°.
Length of an arc
Arc Length (degrees)l = θ360 × 2πr⚠ NOT in the formula booklet — memorise it
Quick example: If θ = 90° and r = 8 cm, then l = 90360 × 2π(8) = 4π ≈ 12.6 cm.
Perimeter problems
For perimeter of a sector or a “remaining shape” question, you usually need to add the arc length plus two radii. Read the question carefully — the perimeter is not the same as the arc length on its own.
Perimeter of a sectorP = arc + 2r = θ360 × 2πr + 2r
Area of a sector
Sector Area (degrees)A = θ360 × πr2⚠ NOT in the formula booklet — memorise it
Quick example: If θ = 90° and r = 8 cm, then A = 90360 × π(8)2 = 16π ≈ 50.3 cm2.
Always include the right unit: arc length is in cm or m; sector area is in cm2 or m2. Examiners deduct marks for missing or wrong units.
Worked examples
WE 1
Length of a pizza crust (basic arc)
A circular pizza of radius 12 cm has had a slice cut from it. The angle of the slice is 38°. Find the length of the outside crust of the slice (the minor arc).
From WE 1, find the perimeter of the remaining pizza (after the 38° slice was cut).
Step 1: Find the major angleθ = 360 − 38 = 322°Step 2: Major arc lengthl = 322360 × 2π(12) = 322π15Step 3: Add the two radii (the cut edges)P = 322π15 + 12 + 12 = 91.4395…P = 91.4 cm (3 s.f.)don’t forget the two radii — only the curve isn’t enough!
WE 3
Area of a sector (Jamie’s blue sector)
Jamie divides a circle of radius 50 cm into two sectors: a minor sector of angle 100° and a major sector of angle 260°. He paints the minor sector blue. Find the blue area.
A sector of a circle has radius 9 cm and an arc length of 6π cm. Find the central angle in degrees.
Step 1: Set up the equation6π = θ360 × 2π(9)Step 2: Simplify the right side6π = θ × 18π360 = θπ20Step 3: Solve for θθ = 6π × 20π = 120θ = 120°π cancels — leave answers exact when you can
💡 Top tips
Always sketch a quick diagram. Even a rough circle with the angle marked helps you spot whether you should use the minor or major angle.
Check whether the question wants minor or major. If the question says “remaining” or “the rest of”, you usually need 360 − θ.
Sector ≠ perimeter ≠ arc length. Read the question to know exactly what’s being asked. Perimeter usually = arc + 2r.
Leave π in your working until the final step. Cancellations often clean up the arithmetic before you reach for the calculator.
Mind the units. Length → cm. Area → cm2. Marks are routinely lost for forgotten or wrong units.
If a question gives you angle and arc length, you can find the radius by rearranging the formula.
For “fraction-of-pizza”-style problems, sanity check your answer: a 90° slice should be ¼ of the full area or circumference.
⚠ Common mistakes
Using the wrong formula. Arc uses 2πr (circumference); sector uses πr2 (area). Always double-check which one you’ve written down.
Forgetting the two radii in perimeter. Perimeter of a sector = arc + 2r, not just the arc.
Plugging in the minor angle when the question wants the major. If the slice is “the rest of the circle”, use 360 − θ.
Squaring the wrong thing. In the area formula, the radius is squared but θ/360 is not.
Mixing units. If radius is in cm but the answer is in mm or m, you’ve made a conversion slip.
Using radians by mistake. The degree formulas have θ360; the radian formulas don’t. Watch which you’re using.
Rounding too early. Keep π or full decimals until the very last step, then round to 3 s.f. (or as the question states).
Master these two formulas and the next two notes (radian measure + radian sector formulas) become almost automatic — they’re the same idea, just dressed up differently.
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