Formula

Area and arc length of a circular sector

Mensuration • Plane Figures • Circle

Formula

A = \frac{θ}{360} π r^{2}, ℓ = \frac{θ}{360} \cdot 2 π r

A sector of a circle of radius

r

whose central angle is

θ

degrees has arc length

\frac{θ}{360} \times 2 π r

and area

\frac{θ}{360} \times π r^{2}

; with

θ

in radians these read

r θ

and

\frac{1}{2} r^{2} θ

How to use it

1Read off the fraction first: a $6 0^{\circ}$ sector is a sixth of the circle, a $9 0^{\circ}$ sector a quarter.
2Several sectors whose angles sum to $36 0^{\circ}$ make one whole circle, which is what collapses the three $6 0^{\circ}$ corners of an equilateral triangle into half a circle.
3The segment cut off by a chord is the sector less the triangle formed by the two radii.

Find the arc length and the area of a

6 0^{\circ}

sector of a circle of radius

6

cm.

Step-by-step

1
$\frac{60}{360} = \frac{1}{6}$
Why: The sector is this fraction of the whole circle.
2
$ℓ = \frac{1}{6} \times 2 π \times 6 = 2 π$ cm
3
$A = \frac{1}{6} \times π \times 6^{2} = 6 π$ cm $^{2}$

Answer

arc

2 π

cm, area

6 π

^{2}

Watch out

The sector is bounded by two radii and the arc, so its perimeter is

ℓ + 2 r

, not

ℓ

alone. Do not confuse the sector with the segment, which is the smaller region cut off by the chord.

Builds on

Also known as

area of a sector, arc length, length of an arc, sector of a circle, minor sector, fraction of a circle