Learn the method, study an example, then explain your reasoning in an exam-style problem. Work on paper before opening the solution and compare each step with the marking guidance.
Arc length, sector perimeter and rolling distance
Choose the correct arc and count every part of the required boundary.
Notes
Before you start: convert degrees to radians and distinguish radius from diameter.
For radius r and an angle θ in radians, the arc length is \(s=r\theta\). For a proper sector, \(0<\theta<2\pi\), its boundary consists of the arc and two radii:
\[P=2r+r\theta=r(2+\theta).\]
At a full circle the radial edges are not part of the outside boundary: its circumference is 2πr, not 2r+2πr. A chord is a straight segment; it cannot replace an arc in these formulae.
A wheel rolling without slipping travels one circumference per revolution. More generally, travelled distance is radius times total swept angle. With constant angular speed ω in rad/s, distance after t seconds is \(r\omega t\). Check that all lengths use compatible units.
Recall later
An arc of length 2.4 cm on a radius of 4 cm subtends \(\displaystyle \theta = \frac{2.4}{4}= 0.6\) rad.
Common mistakes
Giving only \(\displaystyle r\theta\) when the question asks for sector perimeter.
Including two radii in the outside boundary of a full disk.
Using wheel diameter in place of radius in \(\displaystyle r\theta\).
Three worked examples
Example 1
For a sector of radius 6 cm and angle 0.8 rad, find arc length and perimeter.
Show the reasoning
The arc is \(6(0.8)=4.8\) cm. Its complete perimeter is \(4.8+2(6)=16.8\) cm.
Example 2
An arc has length 9 m on a circle of radius 3 m. Find its central angle.
Show the reasoning
\(\displaystyle \theta = \frac{s}{r}= \frac{9}{3}= 3\) rad. The angle is not 3 degrees; the quotient of two compatible lengths defines radians.
Example 3
A wheel of radius 0.4 m rolls five complete revolutions without slipping. Find its travelled distance.
Show the reasoning
The total angle is \(10\pi\) rad, so distance \(=0.4(10\pi)=4\pi\) m, approximately 12.6 m to 3 significant figures.
