Radians, Arcs & Sectors (AA SL)
Radians are the second way of measuring angles, and once you're comfortable converting between them and degrees, they make circle geometry surprisingly clean - arc length and sector area formulas that need no extra conversion factor at all. This topic covers converting degrees to radians and back, finding the length of an arc, the area of a sector (the pie-slice shape), and the area of a segment (the region cut off by a chord).
What the syllabus says
This topic maps onto one point in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| SL3.4 | The circle: radian measure of angles; length of an arc; area of a sector. Radian measure may be expressed as exact multiples of \(\pi\), or as decimals. |
This is a core AA SL syllabus point, examined on Paper 1 and Paper 2. On IB exam papers, radian measure is assumed unless a question states otherwise.
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 a radian?
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. There are \(2\pi\) radians in a full turn, matching the \(360^\circ\) you already know.
e.g. \(\pi\) radians \(=180^\circ\), so \(1\) radian \(=\dfrac{180}{\pi}\approx57.3^\circ\).
How do you convert degrees to radians?
Multiply the number of degrees by \(\dfrac{\pi}{180}\). The result is often left as an exact multiple of \(\pi\) rather than converted to a decimal.
e.g. \(150^\circ\times\dfrac{\pi}{180}=\dfrac{5\pi}{6}\) radians.
What is arc length?
Arc length is the distance along the curved edge of a sector, found using \(s=r\theta\) with \(\theta\) in radians. No conversion factor is needed because that's exactly how a radian is defined.
e.g. A sector with radius \(6\) cm and angle \(0.9\) radians has arc length \(s=6(0.9)=5.4\) 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 \(A=\tfrac12r^2\theta\), again with \(\theta\) in radians.
e.g. A sector with radius \(6\) cm and angle \(0.9\) radians has area \(A=\tfrac12(36)(0.9)=16.2\) cm\(^2\).
What is the area of a segment?
A segment is the region between a chord and its arc - the sector with the triangular piece (formed by the two radii and the chord) cut away. Its area is the sector area minus \(\tfrac12r^2\sin\theta\).
e.g. With \(r=6\), \(\theta=0.9\): triangle area \(=\tfrac12(36)\sin0.9\approx14.1\), so segment area \(\approx16.2-14.1=2.1\) cm\(^2\).
Key formulas
Two core formulas, plus the conversion factor between degrees and radians, cover almost every question on this topic. The tables below summarise them at a glance - the explanations underneath go into more depth on each one.
Formula reference
The arc length and sector area formulas are printed in the formula booklet; degree-radian conversion and the segment-area subtraction are prior knowledge you're expected to apply.
| Formula | Used for | Booklet? |
|---|---|---|
| \(s=r\theta\) | Arc length | ✓ Yes |
| \(A=\tfrac12r^2\theta\) | Sector area | ✓ Yes |
| Radians \(=\) degrees \(\times\dfrac{\pi}{180}\) | Degree-to-radian conversion | Not in booklet - prior knowledge |
| Segment area \(=\tfrac12r^2\theta-\tfrac12r^2\sin\theta\) | Segment area (sector minus triangle) | Not in booklet - derived by subtraction |
Degrees vs radians
Both measure the same angles - the units and the formulas they unlock are different.
| Feature | Degrees | Radians |
|---|---|---|
| Full turn | \(360^\circ\) | \(2\pi\) |
| Straight line | \(180^\circ\) | \(\pi\) |
| Right angle | \(90^\circ\) | \(\dfrac{\pi}{2}\) |
| Best for | Everyday angle measurement, bearings | Arc length/sector area formulas, calculus with trig functions |
Converting between degrees and radians
Both directions use the same conversion factor, just inverted.
Degrees to radians
\[\text{radians} = \text{degrees}\times\dfrac{\pi}{180}\]
Multiply by \(\pi/180\); simplify the fraction to leave an exact multiple of \(\pi\).
Radians to degrees
\[\text{degrees} = \text{radians}\times\dfrac{180}{\pi}\]
Multiply by \(180/\pi\); the \(\pi\)s cancel if the radian value is an exact multiple of \(\pi\).
Arc length and sector area
Both formulas need the angle in radians - using degrees here is the single most common error on this topic.
Arc length
\[s = r\theta\]
Radius times angle in radians - no extra factor needed.
Sector area
\[A = \tfrac12 r^2\theta\]
Half the radius squared, times the angle in radians.
The area of a segment
A segment is what's left of a sector once you remove the triangle formed by the two radii and the chord.
Triangle piece
\[\text{Triangle area} = \tfrac12 r^2\sin\theta\]
The same area formula from sine/cosine rule topics, using the two radii as sides.
Segment area
\[\text{Segment area} = \tfrac12r^2\theta - \tfrac12r^2\sin\theta\]
Sector area minus the triangle area, both using the same \(r\) and \(\theta\).
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
\(60^\circ.\)
(a) Convert it to radians, in terms of \(\pi\).
(b) Convert \(120^\circ\) to radians.
Worked solution
(a) \(60\times\dfrac{\pi}{180}=\dfrac{\pi}{3}.\) M1
\(\dfrac{\pi}{3}.\) A1
(b) \(\dfrac{2\pi}{3}.\) A1
A sector of a circle has radius 9 cm and angle \(\theta=1.2\) radians at the centre.
(a) Find the length of the arc.
(b) Find the area of the sector.
(c) Find the area of the segment bounded by the arc and its chord.
Worked solution
(a) \(s=r\theta=9(1.2)\) M1
\(=10.8\) cm. A1
(b) \(A=\tfrac12 r^2\theta=\tfrac12(81)(1.2)\) M1
\(=48.6\) cm². A1
(c) Triangle area \(=\tfrac12 r^2\sin\theta=\tfrac12(81)\sin1.2\approx37.74\) cm². M1
Segment \(=48.6-37.74\approx10.9\) cm². 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 \(\theta\) in degrees inside \(s=r\theta\) or \(A=\tfrac12r^2\theta\). Both formulas only work when the angle is in radians - if the question gives degrees, convert first.
- Calculator left in the wrong angle mode. \(\sin1.2\) means something completely different depending on whether the calculator thinks that's \(1.2\) radians or \(1.2^\circ\) - check the mode before evaluating.
- Treating a segment as if it were a sector. The segment area needs the triangle area subtracted from the sector area - just using \(\tfrac12r^2\theta\) alone gives the sector, not the segment.
- Rounding \(\pi\) too early. When a question asks for an answer "in terms of \(\pi\)", leave it as an exact multiple of \(\pi\) (like \(\tfrac{5\pi}{6}\)) rather than converting to a decimal.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
The single most common source of lost marks in trig - your answer is only correct if the angle mode matches the question. AA usually uses radians; AI usually uses degrees.
- Check the question: are angles in degrees (°) or radians (\(\pi\), rad)?
- Press MODE, highlight RADIAN or DEGREE on the angle row and press ENTER, then 2nd → MODE to quit.TI-84
- Press ctrl → menu → Settings (or doc Settings) → Angle, and choose Degree or Radian.Nspire
- Press SHIFT → MENU (SET UP) → Angle, then choose Deg or Rad and EXIT.Casio
- Re-enter the calculation after switching - the mode only affects new work.
Tip: If a sin/cos/tan answer looks wildly wrong, the angle mode is almost always the cause.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Radians, arcs and sectors questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
Why does IB Maths use radians instead of degrees?
Radians make the arc length and sector area formulas simple (s = r theta, A = half r squared theta) with no extra conversion factor, and they're essential later for calculus with trigonometric functions. IB exam papers assume radian measure unless a question says otherwise.
How do I convert between degrees and radians?
Multiply degrees by pi/180 to get radians, or multiply radians by 180/pi to get degrees. Radian answers are often left as exact multiples of pi rather than converted to decimals.
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 the region bounded by a chord and an arc - you find its area by subtracting the triangle area (half r squared sin theta) from the sector area.
Can I use my GDC for this topic?
Yes - once your calculator is set to radian mode, it evaluates sin, cos, and tan of radian angles directly, and can also switch a decimal radian answer to an exact multiple of pi. See the GDC guide below for model-specific steps.
Sub-topics
Radians, Arcs & Sectors broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AA SL syllabus unit, in case you want to keep going.