4.9bHigher tier1MA1 · Section 4: Geometry and trigonometry

Arc length and area of a sector

Key learnings

A sector is a fraction of the whole circle
The fraction is $\dfrac{\theta}{360}$ . Arc length $= \dfrac{\theta}{360}\times 2\pi r$ ; sector area $= \dfrac{\theta}{360}\times \pi r^2$ .
Perimeter of a sector includes the two radii
Perimeter $= \text{arc length} + 2r$ — don't forget the straight edges.

Drawn from real examiner reports.

Rearranging the area formula incorrectly

When solving for $r$ or $\theta$ , algebra slips are common; keep the $\dfrac{\theta}{360}$ fraction intact while rearranging.

Confusing sector area with the area of a triangle/segment

Some find a triangle area or subtract values that the question did not ask for.

Write the formula, substitute, then evaluate
Show the substitution before pressing buttons — it earns method marks even if the arithmetic slips.

For a sector of radius $r$ with angle $\theta$ (in degrees):

Both are just the fraction $\dfrac{\theta}{360}$ of the whole circle.

Foundation

What fraction of the circle is a sector?

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