Finding the Angle from the Arc Length

The arc formula $s = r\theta$ runs both ways. Given the arc and radius, the angle is $\theta = \frac{s}{r}$, in radians. Then the sector area follows from $\frac12 r^2\theta$.

A sector of a circle has radius $6$ cm and an arc length of $15$ cm. (i) Find the angle of the sector in radians. [2] (ii) Find the area of the sector. [2]

The working

(i) Rearrange $s = r\theta$ to find $\theta$: $\theta = \frac{s}{r} = \frac{15}{6} = 2.5 \text{ radians} \quad \text{(M1, A1)}$

(ii) Sector area with $r = 6$, $\theta = 2.5$: $A = \frac{1}{2}r^2\theta = \frac{1}{2}(6)^2(2.5) = \frac{1}{2}(36)(2.5) = 45 \text{ cm}^2 \quad \text{(M1, A1)}$

Where the marks are won and lost

• $\theta = \frac{s}{r} = \frac{15}{6} = 2.5$ is in radians, use it directly in the radian area formula. Treating it as degrees (and converting) is wrong.
• The area formula is $\frac12 r^2\theta$ (radians), not $\frac{\theta}{360}\pi r^2$. Match the formula to the angle unit.
• $\frac12 \times 36 \times 2.5 = 45$: keep the arithmetic clean.

Common mistakes

• Treating $2.5$ as degrees and converting before finding the area.
• Using the degree area formula with a radian angle.
• Slips in $\frac12 \times 36 \times 2.5$.

Full method: Arc Length notes. See also Sector Area. Topic home: Circular Measure pillar.