Arc Length and Sector Area (AA SL)
Once an angle is in radians, arc length and sector area both come from short formula-booklet formulas - but exam questions rarely hand you \(r\) and \(\theta\) directly, they make you rearrange to find whichever one is missing. This page covers both formulas, how to rearrange for an unknown radius or angle, and the slip that turns a correct method into a wrong final answer. It's part of the broader Radians, Arcs & Sectors topic.
17 questions on this sub-topic.
The two formulas
Covered under IB syllabus reference SL3.4: the circle - radian measure of angles, length of an arc, area of a sector. Both formulas below are in the formula booklet, and both require \(\theta\) to be in radians.
Arc length
\[s = r\theta\]
Radius times angle in radians - no extra factor needed.
Sector area
\[A=\tfrac12r^2\theta\]
Half the radius squared, times the angle in radians. Rearranges to \(\theta = \tfrac{2A}{r^2}\) if the area is given instead.
Still converting degrees to radians first? See Radians and Conversion. For angle-mode GDC settings, see Radians, Arcs & Sectors.
Worked examples
A sector has radius 5 cm and central angle \(\tfrac{\pi}{4}\) rad.
Find its exact area.
Worked solution
Sector area \(A=\tfrac12 r^2\theta\). M1
\(r=5,\ \theta=\tfrac{\pi}{4}\): \(A=\tfrac12(25)\tfrac{\pi}{4}=\tfrac{25\pi}{8}\approx 9.82\text{ cm}^2.\) A1
A sector of a circle of radius 10 cm has area \(60\) cm².
Find the central angle in radians.
Worked solution
Choose \(A=\tfrac12 r^2\theta\) because we know the area and radius and want \(\theta\). M1
\(60=\tfrac12(10)^2\theta=50\theta\) A1
so \(\theta=\tfrac{60}{50}=1.2\) rad. A1
A pendulum of length 1.2 m swings through an angle of \(25^\circ\).
Find the distance travelled by the tip in one swing, to 3 significant figures
Worked solution
The arc-length formula needs radians: \(25^\circ=25\cdot\tfrac{\pi}{180}=0.4363\) rad. M1
\(s=r\theta=1.2(0.4363)\approx 0.524\text{ m}.\) A1
Common mistakes
- Squaring the radius in the arc length formula. \(s=r\theta\) has no square - that's the sector area formula. Mixing the two up is the single most common error on this sub-topic.
- Forgetting the \(\tfrac12\) in the sector area formula. \(A=\tfrac12r^2\theta\) is easy to write as \(A=r^2\theta\) under pressure, which doubles the answer.
- Rearranging \(A=\tfrac12r^2\theta\) incorrectly for \(\theta\). Dividing by \(r^2\) alone forgets the \(\tfrac12\); the correct rearrangement is \(\theta=\tfrac{2A}{r^2}\).
- Leaving the calculator in degree mode. Both \(s=r\theta\) and \(A=\tfrac12r^2\theta\) only work when \(\theta\) is in radians - if the angle is given in degrees, convert it first (multiply by \(\pi/180\)) or the answer comes out wildly wrong, since neither formula includes a degrees-to-radians step of its own.
Ready to practise properly?
18 arc length and sector area questions, marked instantly like the real exam.
Quick answers
What is the formula for arc length?
\(s = r\theta\), where \(r\) is the radius and \(\theta\) is the angle at the centre in radians. It is in the formula booklet.
What is the formula for sector area?
\(A = \tfrac12 r^2\theta\), where \(\theta\) must be in radians. It is in the formula booklet, and rearranges easily if the area is given and \(\theta\) is unknown.