The idea

Arc length & sector area

Why a sector is just a fraction of the whole circle, arc length and sector area from that one idea, perimeter of a sector (don't forget the radii), and working backwards from an arc or area to the angle.

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Geometry & measures · Circles, arcs & sectors

Arc length & sector area

Why it works

A sector is a fraction of the circle

A sector is a fraction of the circle, and the angle tells you which fraction. A full turn is

360°

, so a sector with angle

\theta

\frac{\theta}{360}

of the whole circle. Everything follows from that single fraction:

\text{arc} = \frac{\theta}{360} \times 2\pi r, \qquad \text{area} = \frac{\theta}{360} \times \pi r^2

There is nothing else to memorise — if you can remember the circumference

2\pi r

and the area

\pi r^2

, you can rebuild both. A quick check: a

90°

sector should give exactly a quarter of the circle.

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