IB Maths AA SLTopic 3 — Geometry & TrigPaper 1 & 2~7 min read
Arcs & Sectors Using Radians
When the angle is in radians, the arc length and sector area formulas get dramatically cleaner. No more θ360, no more 2π in front. Just two compact equations — both in your formula booklet.
📘 What you need to know
Arc length:l = rθ
Sector area:A = 12r2θ
θ must be in radians for these formulas to work — convert first if you’re given degrees.
Both formulas are in the IB formula booklet (geometry & trigonometry section).
Same idea as the degree formulas — they’re just much cleaner because radians are designed to make this easy.
Why radians win for circle problems
If you came from the previous note (Arcs & Sectors Using Degrees), you’ll appreciate this. Look what happens to the formulas when we switch to radians:
🎉 Radians make the formulas way simpler
θ360 × 2πr→l = rθ
θ360 × πr2→A = 12r2θ
🤔 Why are the radian formulas simpler?
Remember: a radian is defined as the angle whose arc length equals the radius. So 1 radian gives an arc of length r; 2 radians gives an arc of length 2r; θ radians gives an arc of length rθ. The formula l = rθ is literally the definition of a radian. No conversion factors needed.
The two formulas (memorise the structure)
📏
Arc Length
l = rθ
r = radius θ = angle in radians
✓ in formula booklet
🍕
Sector Area
A = 12r2θ
r = radius θ = angle in radians
✓ in formula booklet
A sector of a circle — labelled in radians
Length of an arc
Arc Length (radians)l = rθ
✓ in formula booklet
Quick example: A circle of radius 10 cm has a sector with angle π5. Arc length = rθ = 10 × π5 = 2π ≈ 6.28 cm.
Perimeter of a sector
Same idea as before — the perimeter is the arc plus the two radii:
Sector PerimeterP = rθ + 2r
Area of a sector
Sector Area (radians)A = 12r2θ
✓ in formula booklet
Quick example: Same circle as above (radius 10 cm, angle π5). Area = 12(10)2(π5) = 10π ≈ 31.4 cm2.
Notice both formulas are in the IB formula booklet, so you don’t strictly need to memorise them. But you should — flipping pages mid-paper costs valuable time.
Degrees vs radians — full comparison
Same problem, two ways. The radian column is shorter every time:
Quantity
Using degrees
Using radians
Arc length
θ360 × 2πr
rθ
Sector area
θ360 × πr2
12r2θ
In formula booklet?
No
Yes ✓
Need to memorise?
Yes
Optional
Bottom line: if a question gives you the angle in degrees, your first move is usually to convert it to radians, then use the simpler formula.
Worked examples
WE 1
Find the arc length
A sector of a circle has radius 8 cm and central angle 3π4. Find the arc length.
Step 1: Pick the formulal = rθStep 2: Substitute r = 8, θ = 3π/4l = 8 × 3π4Step 3: Simplify= 24π4 = 6πl = 6π ≈ 18.8 cmleave answer as 6π unless decimal is requested
WE 2
Find the sector area
A sector of a circle has radius 6 cm and central angle π3. Find the area.
Step 1: Pick the formulaA = 12r2θStep 2: Substitute r = 6, θ = π/3A = 12(6)2(π3)Step 3: Compute= 12 × 36 × π3 = 36π6 = 6πA = 6π ≈ 18.8 cm2don’t forget the units — cm² for area!
WE 3
Cake slice — area and perimeter (SME-style)
A slice of cake forms a sector with angle π6 and radius 7 cm. Find:
(a) the area of the surface of the slice (b) the perimeter of the slice
(a) AreaA = 12(7)2(π6) = 49π12A ≈ 12.8 cm2 (3 s.f.)(b) Perimeter = arc + 2r
Arc: l = 7(π6) = 7π6
Perimeter: P = 7π6 + 2(7) = 7π6 + 14P ≈ 17.7 cm (3 s.f.)always include the 2 radii in the perimeter!
WE 4
Reverse — find the radius
A sector has arc length 15 cm and central angle 1.2 radians. Find the radius.
Step 1: Use the arc length formulal = rθStep 2: Substitute and rearrange15 = r(1.2)r = 151.2Step 3: Computer = 12.5 cmno need to convert — angle already in radians
WE 5
Convert first, then use the formula
A sector has radius 9 cm and central angle 80°. Find the area, giving your answer in cm2.
Step 1: Convert 80° to radians80 × π180 = 80π180 = 4π9Step 2: Apply the area formulaA = 12(9)2(4π9)Step 3: Compute= 12 × 81 × 4π9 = 18πA = 18π ≈ 56.5 cm2always convert to radians BEFORE using the formula
💡 Top tips
Always check the angle is in radians before using these formulas. If degrees are given, convert first using ×π180.
Both formulas are in the formula booklet — but it’s faster to know them by heart.
Leave answers as multiples of π in your working. Only convert to a decimal at the end if asked.
Don’t forget units. Arc length is cm (or m); area is cm² (or m²).
For perimeters, always remember to add the two straight radii to the curved arc.
If a problem mixes degrees and radians, convert everything into radians first — these formulas only work in radians.
Sanity check: for a full circle (θ = 2π), l = r(2π) = 2πr ✓ and A = ½r²(2π) = πr² ✓ — both formulas match the standard circle formulas when θ = 2π.
⚠ Common mistakes
Using radians formulas with the angle in degrees. The biggest mistake — putting θ = 60 instead of θ = π/3. Always check.
Forgetting the ½ in the area formula. The sector area is ½r²θ, not r²θ.
Squaring θ instead of r. Only the radius is squared in the area formula.
Mixing up arc length and sector area formulas. Length uses one factor of r; area uses two.
Wrong calculator mode. If you compute sin or cos in a problem with radians, make sure your calculator is in RAD mode.
Forgetting to add the radii for perimeter. The arc on its own is not the perimeter of a sector.
Wrong units in the final answer. cm for arc length, cm² for area — easy marks lost otherwise.
That’s the entire Geometry Toolkit done. You can now find midpoints, distances, gradients, arc lengths, sector areas, and convert angles in either direction. These are the foundational tools that show up everywhere in trig, vectors, and beyond.
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