Finding arc length and area of a sector

A sector is the 'pizza slice' region of a circle bounded by two radii and the arc between them. In the Circular Measure chapter you are asked to find the length of that curved edge, the arc length, and the area enclosed by the sector.

Both have short formulas, but they only work when the angle at the centre is measured in radians.

This method also handles related quantities: the perimeter of a sector (the arc plus the two straight radii), and the area of a segment (a sector with a triangle removed). Circular measure questions often combine these with the sine and cosine rules, so a secure grip on $s=r\theta$ and $A=\frac{1}{2}r^{2}\theta$ is the foundation of the whole chapter.

Because the marking is analytic, showing each substitution clearly earns method marks even if the final arithmetic slips.

Reach for these formulas whenever a question shows a circle or part of a circle with a marked centre, a radius, and an angle, and asks for a curved length or an area of a slice. Words such as 'arc', 'sector', 'perimeter of the sector' or 'area of the shaded region' are the signals.

Before you substitute, always check the units of the angle. If it is given in radians (often as a decimal like $0.75$ or a multiple of $\pi$), you are ready.

If it is given in degrees, convert first using $\theta_{\text{rad}}=\theta_{\text{deg}}\times\frac{\pi}{180}$. Using degrees directly in $s=r\theta$ is the most common way to lose marks here, so make the radian check an automatic first step.

1. 1

   Check the angle

   Confirm $\theta$ is in radians. If it is in degrees, convert with $\theta_{\text{rad}}=\theta_{\text{deg}}\times\frac{\pi}{180}$.

2. 2

   List r and θ

   Read the radius $r$ and the angle $\theta$ from the figure, with their units.

3. 3

   Arc length

   Substitute into $s=r\theta$.

4. 4

   Perimeter, if asked

   Add the two radii to the arc: perimeter $=s+2r$.

5. 5

   Sector area

   Substitute into $A=\frac{1}{2}r^{2}\theta$.

6. 6

   State units

   Give lengths in cm and areas in cm² (or the units used in the question).

• Leaving the angle in degrees and substituting straight into $s=r\theta$ or $A=\frac{1}{2}r^{2}\theta$. Convert to radians first.
• Forgetting the $\frac{1}{2}$ in the area formula, or squaring the wrong quantity, it is $r^{2}$, not $(r\theta)^{2}$.
• Confusing the perimeter of a sector with the arc length: the perimeter also includes the two radii, so add $2r$.
• Squaring the radius but forgetting to multiply by $\theta$, or multiplying by $\theta$ but not squaring $r$.
• Writing the area in cm instead of cm², or the length in cm², always match the units to the quantity.

The step students most often get wrong is the radian check: they read $60^{\circ}$ or $45^{\circ}$, substitute it as if it were radians, and every later answer is wrong. In a one-to-one lesson our teachers build the habit of circling the angle and asking 'radians or degrees?'

before anything else, so the conversion becomes automatic. We also keep arc length, perimeter and area clearly separated so you never mix the formulas.

Because your working is shown line by line, each substitution earns its method mark. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

To see how we teach circular measure, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.