Sector Area, and Finding the Angle from the Perimeter

This question tests whether you can use the sector formulae in reverse: work back from the perimeter to the angle, then forward to the area. Everything hinges on remembering that perimeter $= r\theta + 2r$, so the two radii come off first.

A sector of a circle has radius $6$ cm and a perimeter of $20$ cm. (i) Find the angle of the sector in radians. [3] (ii) Find the area of the sector. [2]

The working

(i) The perimeter is arc plus two radii: $P = r\theta + 2r$. Substitute $P = 20$, $r = 6$: $20 = 6\theta + 2(6) = 6\theta + 12 \quad \text{(M1)}$

Solve for $\theta$: $6\theta = 8 \;\Rightarrow\; \theta = \frac{8}{6} = \frac{4}{3} \approx 1.33 \text{ radians} \quad \text{(M1, A1)}$

(ii) Sector area uses $A = \frac{1}{2}r^2\theta$: $A = \frac{1}{2}(6)^2 \times \frac{4}{3} = \frac{1}{2}(36)\times\frac{4}{3} = 18 \times \frac{4}{3} = 24 \text{ cm}^2 \quad \text{(M1, A1)}$

Where the marks are won and lost

• Subtract both radii: $P - 2r = r\theta$. Forgetting the $2r$ (using $20 = 6\theta$) gives $\theta \approx 3.33$ and a wrong area.
• Keeping $\theta = \frac{4}{3}$ as an exact fraction avoids rounding drift into part (ii). $\frac12 \times 36 \times \frac43 = 24$ is exact.
• $A = \frac12 r^2\theta$ (radians), not $\frac{\theta}{360}\pi r^2$. Match the formula to the angle unit.

Common mistakes

• Solving $20 = 6\theta$ (omitting the two radii).
• Using the degree area formula with a radian angle.
• Rounding $\theta$ to $1.3$ early and losing accuracy in the area.

Full method: Sector Area notes. See also Problem-Solving with Circular Measure. Topic home: Circular Measure pillar.