Radians and Sectors (AA HL)
Radians measure angles as a fraction of the circle's own radius rather than a fixed 360-degree scale, and once you're working in radians, arc length and sector area both come from short, direct formulas. This page covers those formulas and the extra step needed for a segment, with worked examples and the mode mistake that trips up most students. It's part of the broader Triangles & Circular Functions topic.
10 questions on this sub-topic.
The key formulas
Covered under IB syllabus reference SL3.4: the circle in radian measure - length of an arc and area of a sector. Both formulas below are in the formula booklet; the segment formula is a short combination of the two that you're expected to build yourself.
Arc length
\(s = r\theta\)
\(\theta\) must be in radians. Use when a question asks for the length of a curved arc, not the straight chord across it.
Sector area
\(A = \tfrac12 r^2\theta\)
The "pizza slice" area bounded by two radii and the arc between them. Again, \(\theta\) is in radians.
Segment area
\(A = \tfrac12 r^2(\theta - \sin\theta)\)
A segment is the sector minus the triangle formed by the two radii and the chord: sector area \(\tfrac12r^2\theta\) minus triangle area \(\tfrac12r^2\sin\theta\).
Want the sine rule, cosine rule and triangle area formulas as well? See Triangles & Circular Functions.
Worked examples
A sector has radius \(r=6\) cm and angle \(\theta=\dfrac\pi3.\) Find the arc length.
Worked solution
\(s = r\theta = 6\cdot\dfrac\pi3\) M1
\(= 2\pi \approx 6.28\) cm. A1
A chord subtends an angle of \(1.6\) rad at the centre of a circle of radius \(10\) cm.
Find the area of the minor segment.
Worked solution
area \(= \tfrac12 r^2(\theta - \sin\theta)=\tfrac12(100)(1.6 - \sin1.6).\) M1
\(= 50(1.6 - 0.9996).\) A1
\(50(0.6004)\approx 30.0\) cm². A1
Convert to radians (exact).
(a) \(60^\circ\)
(b) \(225^\circ\)
Worked solution
(a) Multiply by \(\tfrac{\pi}{180}\): M1
\(60\times\tfrac{\pi}{180}=\tfrac{\pi}{3}.\) A1
(b) \(225\times\tfrac{\pi}{180}=\tfrac{5\pi}{4}.\) A1
Common mistakes
- Mixing up degree and radian mode. Triangle problems are usually in degrees, but circular function models are usually in radians - check your GDC's mode before evaluating anything.
- Forgetting the segment is not the sector. The sector includes the two straight radii; the segment is bounded by the chord instead, so it needs the extra \(-\sin\theta\) term subtracted off.
- Plugging degrees straight into \(s=r\theta\) or \(A=\tfrac12r^2\theta\). Both formulas only work with \(\theta\) in radians - convert first if the angle is given in degrees.
- Confusing the sector's perimeter with its arc length. A sector's perimeter is the arc plus the two straight radii, \(r\theta+2r\) - not just the arc length \(r\theta\) on its own.
Ready to practise properly?
13 radians-and-sectors questions, marked instantly like the real exam.
Quick answers
What is the formula for arc length in radians?
\(s = r\theta\), where \(r\) is the radius and \(\theta\) is the angle at the centre measured in radians.
What is the formula for the area of a sector?
\(A = \tfrac12 r^2\theta\), with \(\theta\) in radians. Both this and the arc length formula are given in the formula booklet.