Radians, Arcs and Sectors (AI HL)
At HL, angles at the centre of a circle are measured in radians rather than degrees, and that single change simplifies both the arc-length and sector-area formulas. This page covers the two radian formulas, where they come from, and the mistakes that cost marks when students default back to the degree versions from SL. It's part of the broader Geometry & Trigonometry topic.
14 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference AHL3.7: the definition of a radian, and using radians to find arc length and sector area. Both formulas are in the formula booklet, but only work correctly when the angle is in radians.
Arc length
\(s = r\theta\)
\(\theta\) must be in radians. Use this when the question gives (or asks for) the length of the curved edge of a sector.
Sector area
\(A = \dfrac{1}{2}r^2\theta\)
Also needs \(\theta\) in radians. This replaces the degree-based \(\tfrac{\theta}{360}\times\pi r^2\) formula you may have seen at SL.
Want the full topic overview, syllabus table and GDC guidance? See Geometry & Trigonometry (including using your GDC).
Worked examples
A circle has radius 9 cm.
Find the length of an arc subtending an angle of \(\tfrac{2\pi}{3}\) rad at the centre.
Worked solution
Arc length \(s=r\theta\), with \(\theta\) in radians (it is). M1
\(s=9\times\tfrac{2\pi}{3}=6\pi\approx 18.8\text{ cm}.\) A1
A sector has radius 10 cm and angle \(1.2\) radians.
Find the arc length and sector area.
Worked solution
Arc \(= r\theta = 10(1.2)\) M1 \(= 12\) cm. A1
Area \(= \tfrac12 r^2\theta\) M1 \(= \tfrac12(100)(1.2) = 60\) cm². A1
Common mistakes
- Reaching for the degree formula out of habit. Writing \(\tfrac{\theta}{360}\times 2\pi r\) or \(\tfrac{\theta}{360}\times \pi r^2\) when \(\theta\) is already in radians double-converts the angle and gives a nonsense answer.
- Leaving the calculator in degree mode. If \(\theta\) is given as a decimal like \(1.2\) with no \(\pi\) or degree symbol, it's radians - check the GDC's angle mode before evaluating \(\sin\), \(\cos\) or \(\tan\) anywhere else in the same question.
- Dropping the \(\tfrac12\) in the area formula. Arc length is \(r\theta\) but sector area is \(\tfrac12 r^2\theta\) - mixing the two up (or forgetting the square on \(r\)) is an easy way to lose an A1.
- 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 the arc length \(r\theta\) on its own - forgetting the extra \(2r\) is an easy way to lose a mark on an otherwise fully correct calculation, especially when a question asks for the perimeter straight after the arc length.
Ready to practise properly?
13 radians, arcs and sectors questions, marked instantly like the real exam.
Quick answers
What is the formula for the length of an arc in radians?
\(s = r\theta\), where \(r\) is the radius and \(\theta\) is the angle subtended at the centre, measured in radians.
What is the formula for the area of a sector in radians?
\(A = \tfrac12 r^2\theta\), with \(\theta\) in radians - not the degree version \(\tfrac{\theta}{360}\times\pi r^2\) you may have used at SL.