Arc Length and the Perimeter of a Sector

Circular measure is one of the most reliable mark sources in 0606, provided you keep the angle in radians and remember what a sector’s perimeter actually includes. The formulae are short: arc $= r\theta$, and perimeter adds the two straight radii.

A sector of a circle has radius $8$ cm and subtends an angle of $1.2$ radians at the centre. (i) Find the length of the arc. [2] (ii) Find the perimeter of the sector. [2]

The working

(i) Arc length uses $s = r\theta$ (with $\theta$ in radians): $s = 8 \times 1.2 = 9.6 \text{ cm} \quad \text{(M1, A1)}$

(ii) The perimeter is the arc plus the two radii that bound the sector: $P = s + 2r = 9.6 + 2(8) = 9.6 + 16 = 25.6 \text{ cm} \quad \text{(M1, A1)}$

Where the marks are won and lost

• $s = r\theta$ requires radians. If an angle were given in degrees, you would convert first ($\times \frac{\pi}{180}$); here $1.2$ is already in radians.
• The perimeter includes $2r$. Quoting $9.6$ cm as the perimeter (arc only) is the classic dropped mark, the two radii are part of the boundary.
• Units throughout (cm). A bare number can lose the final accuracy mark on a measures question.

Common mistakes

• Giving the arc length as the perimeter.
• Using the degree formula $\frac{\theta}{360} \times 2\pi r$ with $\theta = 1.2$ (mixing radian and degree methods).
• Forgetting units.

Full method: Arc Length notes. Topic home: Circular Measure pillar.