Arcs & Sectors (AI SL)
This topic is about the parts of a circle that aren't the whole thing: the curved arc length around part of the circumference, and the pie-slice sector area it cuts off. Both formulas are simple fractions of the full circle, scaled by the central angle, and questions often build them into a chord, segment or real-world "slice" context.
What the syllabus says
This topic maps onto one point in the official IB Applications & Interpretation syllabus.
| Code | Syllabus content |
|---|---|
| SL3.4 | The circle: length of an arc; area of a sector. Radians are not required at SL. |
Chords and segments build directly on this content, combining a sector with the triangle formed by its two radii.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is arc length?
Arc length is the distance along the curved part of a circle between two points, for a given central angle. It's the same fraction of the full circumference \(2\pi r\) as the angle is of a full turn.
e.g. For \(r=15\) cm and \(\theta=80^\circ\): arc \(=\dfrac{80}{360}\times2\pi(15)\approx20.9\) cm.
What is the area of a sector?
A sector is the pie-slice region bounded by two radii and an arc. Its area is the same fraction of the full circle's area \(\pi r^2\) as the central angle is of \(360^\circ\).
e.g. For \(r=11\) cm and \(\theta=75^\circ\): area \(=\dfrac{75}{360}\times\pi(11)^2\approx79.2\) cm\(^2\).
What is a chord?
A chord is the straight line joining two points on a circle - unlike a radius, it doesn't pass through the centre. The two radii and a chord together form an isosceles triangle, so \(\text{chord}=2r\sin\left(\tfrac{\theta}{2}\right)\).
e.g. For \(r=12\) cm and \(\theta=100^\circ\): chord \(=2(12)\sin50^\circ\approx18.4\) cm.
What is a segment?
A segment is the region between a chord and its arc - what's left of a sector once the triangle (formed by the two radii and the chord) is removed. Segment area = sector area − triangle area.
e.g. For \(r=12\), \(\theta=100^\circ\): sector \(\approx125.7\), triangle \(=\tfrac12(12)^2\sin100^\circ\approx70.9\), so segment \(\approx54.8\) cm\(^2\).
What is the perimeter of a sector?
The boundary of a sector has three parts: the curved arc, and the two straight radii on either side. Perimeter = arc length \(+\ 2r\) - it's easy to forget the two radii and quote only the arc.
e.g. For \(r=15\) cm and arc \(\approx20.9\) cm: perimeter \(\approx20.9+2(15)=50.9\) cm.
Key formulas
Two core formulas, plus two that build on them, cover every question on this topic. The two tables below summarise them at a glance - the explanations underneath go into more depth on each one.
Formula reference
Arc length and sector area are on the official formula booklet in exactly this degree-based form; chord length and segment area are derived from them rather than listed separately.
| Formula | Used for | Booklet? |
|---|---|---|
| \(l=\dfrac{\theta}{360}\times2\pi r\) | Arc length | ✓ Yes |
| \(A=\dfrac{\theta}{360}\times\pi r^2\) | Area of a sector | ✓ Yes |
| \(\text{chord}=2r\sin\left(\tfrac{\theta}{2}\right)\) | Chord length | Not in the formula booklet - prior knowledge |
| \(\text{segment}=\text{sector}-\tfrac12 r^2\sin\theta\) | Area of a segment | Not in the formula booklet - prior knowledge |
Sector vs segment
These two regions are easy to confuse, but the boundary that defines each one is different.
| Feature | Sector | Segment |
|---|---|---|
| Bounded by | Two radii and an arc | A chord and an arc |
| Shape | Pie-slice | What's left after the "pie-slice" triangle is cut off |
| Area formula | \(\dfrac{\theta}{360}\times\pi r^2\) | Sector area \(-\) triangle area |
| Perimeter includes | Arc \(+\) two radii | Arc \(+\) one chord |
Sector formulas
Both formulas scale the whole-circle result by the same fraction, \(\dfrac{\theta}{360}\).
Arc length
\[l=\dfrac{\theta}{360}\times2\pi r\]
The fraction of the circumference swept out by the central angle.
✓ In the formula bookletSector area
\[A=\dfrac{\theta}{360}\times\pi r^2\]
The fraction of the circle's total area swept out by the central angle.
✓ In the formula bookletPerimeter of a sector
\[P=l+2r\]
Arc length plus the two straight radii that bound it.
Not in the formula booklet - combine two resultsCircle relationships
Chords and segments bring the triangle area formula from triangle trigonometry into a circle context.
Chord length
\[\text{chord}=2r\sin\left(\tfrac{\theta}{2}\right)\]
Comes from splitting the isosceles triangle (two radii, one chord) into two right triangles.
Not in the formula booklet - prior knowledgeSegment area
\[\text{segment}=\dfrac{\theta}{360}\pi r^2-\tfrac12 r^2\sin\theta\]
Sector area minus the triangle formed by the two radii and the chord.
Not in the formula booklet - combine two resultsSanity check at \(\theta=360^\circ\)
Substituting \(\theta=360\) into the arc and sector formulas should return the full circumference \(2\pi r\) and full area \(\pi r^2\) - a quick way to check you haven't mixed up the formulas.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A sector has radius \(11\) cm and angle \(75^\circ\).
Find (a) the arc length; (b) the area, each to 3 significant figures.
(c) Find the perimeter of the sector.
Worked solution
(a) \(\dfrac{75}{360}(2\pi)(11)\) M1
\(\approx14.4\) cm. A1
(b) \(\dfrac{75}{360}\pi(11)^2\) M1
\(\approx79.2\) cm\(^2\). A1
(c) Perimeter \(=\text{arc}+2r\approx14.4+22\) M1
\(=36.4\) cm. A1
A sector of radius \(12\) cm has angle \(100^\circ.\)
(a) Find the area of the segment, to 3 significant figures.
(b) Find the perimeter of the segment (arc + chord).
(c) Find the ratio of the segment area to the sector area.
Worked solution
(a) Sector \(\dfrac{100}{360}\pi(144)\approx125.7;\) triangle \(\tfrac12(144)\sin100^\circ\approx70.9;\) segment \(\approx54.8\) cm\(^2\). M1 A1
(b) Chord \(=2(12)\sin50^\circ\approx18.4;\) arc \(\approx20.9;\) perimeter \(\approx39.3\) cm. M1 A1
(c) \(\dfrac{54.8}{125.7}\) M1
\(\approx0.436.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Using \(\dfrac{\theta}{2\pi}\) instead of \(\dfrac{\theta}{360}\). At SL these formulas use degrees, not radians - dividing by \(2\pi\) is the radian version and gives a completely wrong fraction here.
- Confusing sector area with segment area. A sector is the full pie-slice; a segment is what's left after the triangle (two radii and the chord) is subtracted - forgetting that subtraction is the most common error on segment questions.
- Quoting only the arc length as the perimeter. A sector's perimeter needs the arc plus both straight radii; a segment's perimeter needs the arc plus the chord - leaving either part out under-counts the boundary.
- Using \(\theta\) instead of \(\tfrac{\theta}{2}\) in the chord formula. The chord formula \(2r\sin(\theta/2)\) comes from splitting the isosceles triangle in half - using the full angle instead of half of it gives the wrong chord length.
Using your GDC
There's no dedicated calculator routine for arcs and sectors - once you've written down the correct formula, it's a single evaluation on the home screen, so the calculator work itself is straightforward.
Type the whole expression - including \(\pi\) and the angle fraction - into your calculator in one line rather than computing the fraction, then \(\pi r\) or \(\pi r^2\), separately, since rounding between steps can shift a 3 s.f. answer. Double-check your calculator's angle mode is set to degrees: none of this content uses radians at SL, but a machine left in radian mode will still evaluate \(\sin\) or other trig terms wrong if a question also involves a chord or triangle area. For a segment question, work out and store the sector area and the triangle area separately before subtracting, so you can see both intermediate values if you need to check your working.
See the full GDC guide for model-specific button sequences across other topics.
Ready to practise properly?
Arcs and sectors questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between a sector and a segment?
A sector is the pie-slice region bounded by two radii and an arc. A segment is bounded by a chord and an arc instead - it's what's left of a sector once you cut off the triangle formed by the two radii and the chord.
Do I need radians for this topic?
No. The IB syllabus explicitly states that radians are not required at SL for the circle topic - arc length and sector area are both calculated using the angle as a fraction of 360 degrees.
How do I find the perimeter of a sector?
Add the arc length to the two straight radii: perimeter = arc length + 2r. It's easy to forget the two radii and just quote the arc length, which only gives part of the boundary.
Is the chord length formula given in the booklet?
No - chord length and segment area aren't listed as standalone formulas in the booklet, so it's worth being comfortable deriving them (chord length uses the isosceles triangle formed by the two radii; segment area is sector area minus triangle area).
Sub-topics
Arcs & Sectors broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AI SL syllabus unit, in case you want to keep going.