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.

Geometry & measures · Circles, arcs & sectors

Arc length & sector area

Why it works

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 length} = \frac{\theta}{360} \times 2\pi r, \qquad \text{sector 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.

Arc length uses the circumference; sector area uses the area. Mixing them up is the standard error. The units help: an arc is a length (cm), a sector is an area (cm²).

Perimeter of a sector ≠ arc length. The boundary of a sector is the curved arc plus the two straight radii:

\text{perimeter} = \frac{\theta}{360} \times 2\pi r + 2r.

A question asking for "the perimeter of the sector" is testing exactly this — the

+2r

is where the mark is.

Working backwards. If you're given the arc length or the sector area and asked for

\theta

r

, set up the same equation and solve. For example, arc

= 12

cm with

r = 9

cm gives

\frac{\theta}{360} \times 18\pi = 12

, so

\theta = \frac{12 \times 360}{18\pi}

.

A segment is the sector minus the triangle. The region between a chord and the arc is what's left when you cut the triangle (two radii and the chord) off the sector:

\text{segment} = \frac{\theta}{360}\pi r^2 - \tfrac{1}{2}r^2\sin\theta.

Exact or rounded. On the non-calculator paper leave answers in terms of

\pi

; on calculator papers keep the full value until the end.