Arc Length and Sector Area (AI SL)
Both an arc and a sector are defined by the same fraction: whatever proportion of \(360^\circ\) the central angle takes up, that's the proportion of the circumference (for the arc) or the total area (for the sector) that you get. Once that fraction is set up correctly, both formulas are direct substitution. This page covers the two formulas with worked examples and the mistakes that lose the most marks. It's part of the broader Arcs & Sectors topic.
40 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL3.4: the length of an arc and the area of a sector. Radians are not required at SL - the central angle is always used as a fraction of \(360^\circ\).
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 bookletNeed the full syllabus wording and formula-booklet reference table? See Arcs & Sectors.
Worked examples
Find the length of an arc of a circle of radius 10 cm that subtends an angle of \(72^\circ\) at the centre.
Worked solution
Arc \(=\dfrac{\theta}{360}\times 2\pi r.\) M1
\(=\dfrac{72}{360}\times 2\pi(10).\) A1
\(=4\pi\approx12.6\) cm. A1
An arc of length 15 cm lies on a circle of radius 12 cm.
Find the angle it subtends at the centre.
Worked solution
\(15=\dfrac{\theta}{360}\times 2\pi(12).\) M1
\(\theta=\dfrac{15\times360}{24\pi}.\) M1
\(\theta\approx71.6^\circ.\) A1
A wheel of radius \(30\) cm makes 100 revolutions. Find the distance travelled, in metres, to 3 significant figures
Worked solution
\(100\times2\pi(0.3)\approx188\) m. M1
\(\approx188\) m. A1
Common mistakes
- Quoting only the arc length as the perimeter. A sector's perimeter needs the arc plus both straight radii - leaving either part out under-counts the boundary.
- Confusing sector area with segment area. A sector is the full pie-slice; a segment is what's left after the triangle formed by the two radii and the chord is subtracted - don't apply the sector formula when the question actually asks for a segment.
- Treating \(\theta\) as already a fraction of the circle. \(\theta\) must be divided by \(360\) first - substituting the raw degree value straight into \(2\pi r\) or \(\pi r^2\) without that division gives a wildly oversized answer.
Ready to practise properly?
43 arc-and-sector questions, marked instantly like the real exam.
Quick answers
What is the formula for arc length?
\(l=\dfrac{\theta}{360}\times2\pi r\), where \(\theta\) is the central angle in degrees and \(r\) is the radius. Radians aren't required at SL - \(\theta\) is always a fraction of \(360^\circ\).
What is the formula for the area of a sector?
\(A=\dfrac{\theta}{360}\times\pi r^2\), the fraction of the circle's total area swept out by the central angle. Both this and the arc length formula are in the formula booklet - see the GDC pointers on the Arcs & Sectors page for calculator setup.