Arc Length

For a sector of radius $r$ and angle $\theta$ in radians:

$s = r\theta$

That’s the whole formula, the marks are in using it with unit discipline and assembling it correctly into perimeters.

Forward, backward, sideways

The exam runs the formula in all directions:

• Forward: $r = 8$, $\theta = 1.5$ $\to$ arc $= 12$ cm
• Backward for $\theta$: arc 10 cm, radius 4 cm $\to$ $\theta = \frac{10}{4} =$ 2.5 radians (no conversion needed, the formula’s output is radians)
• Backward for $r$: arc and angle given, $r = \frac{s}{\theta}$
• Inside an equation: “the perimeter of the sector is 20 cm” $\to$ $r\theta + 2r = 20$, often paired with an area condition to give simultaneous equations in $r$ and $\theta$

If the angle arrives in degrees, convert first, $s = r\theta$ fed degrees is the subtopic’s defining error.

Perimeter: arc plus the straight edges

The perimeter of a sector is $s + 2r$, the arc and both radii. Forgetting the $2r$ is a one-mark leak that examiners report every session. In composite figures, walk the boundary deliberately: which pieces are arcs (use $r\theta$, each with its own $r$ and $\theta$), which are straight (radii, chords)? For a segment, the perimeter is arc + chord, with chord $= 2r \sin(\theta/2)$, derived from the isosceles triangle, as set up in the topic notes.

A sector of radius 6 cm has perimeter 21 cm. Find $\theta$. $r\theta + 2r = 21$ $\to$ $6\theta + 12 = 21$ $\to$ $\theta =$ 1.5 radians The model line "$r\theta + 2r = 21$" is the M mark; the arithmetic is the A.

Exactness and sense-checks

Paper 1 arcs come out clean ($r = 6$, $\theta = \frac{\pi}{3}$ $\to$ arc $= 2\pi$). Sense-check magnitudes: an arc subtending about 1 radian is about one radius long; an arc shorter than expected usually means a degrees-in-radians slip somewhere upstream.

Common mistakes

• Degrees fed into $s = r\theta$
• Perimeter missing the two radii
• Composite boundaries walked carelessly (a chord counted as an arc, an internal radius counted at all)
• $\theta$ “converted” after coming out of $s/\theta$, it’s already in radians
• Half-angle slips in the chord formula

Full topic context: Circular Measure notes · the area partner: sector area.