Arc Length and Area of a Sector

Arc Length and Area of a Sector

A sector of a circle is the region enclosed between two radii and the arc they cut off. Both the length of that arc and the area of that sector are directly proportional to the size of the central angle $\theta$ (measured in radian) that the two radii make -- this section derives the exact formulas from that proportionality.

Area of a sector

The area $A$ of a sector is in the same proportion to the area of the whole circle as its central angle $\theta$ (in radian) is to one full rotation, $2\pi$:

Length of an arc

Similarly, the arc length $S$ of a sector is in the same proportion to the circumference of the whole circle as $\theta$ is to $2\pi$:

Both formulas require $\theta$ to be in radian -- a degree measure must first be converted using $\theta^{c} = \theta°\times\frac{\pi}{180}$ before either formula can be applied. Note also that $A = \frac{1}{2}rS$, since $S = r\theta$.

Solved Examples

Example 1 (arc length from diameter). A circle has diameter $14$ cm; find the arc length subtending $54°$ at the centre.

Radius $r=7$ cm. $\theta = 54\times\frac{\pi}{180}=\frac{3\pi}{10}$ radian. $s=r\theta = 7\times\frac{3\pi}{10} = \frac{21\pi}{10}$, and with $\pi\approx\frac{22}{7}$ this is $\frac{21}{10}\times\frac{22}{7} = \frac{66}{10}=6.6$ cm.

Example 2 (area between an arc and its chord). In a circle of radius $12$ cm, an arc $PQ$ subtends $30°$ at the centre; find the area enclosed between arc $PQ$ and chord $PQ$.

$\theta = 30\times\frac{\pi}{180}=\frac{\pi}{6}$ radian. Sector area $=\frac12 r^2\theta = \frac12(12)(12)\frac{\pi}{6}=12\pi$ sq cm. Drawing the perpendicular from $Q$ to $OP$, the height of $\triangle OPQ$ is $QR = 12\sin30° = 6$ cm, so its area is $\frac12\times12\times6=36$ sq cm. Required area $= 12\pi - 36 = 12(\pi-3)$ sq cm.

Example 3 (arc length and sector area, from the circle's total area). A circle has area $225\pi$ sq cm; find the arc length and sector area for a central angle of $120°$.

$\pi r^2 = 225\pi \Rightarrow r = 15$ cm. $\theta = 120\times\frac{\pi}{180}=\frac{2\pi}{3}$ radian. $s=r\theta = 15\times\frac{2\pi}{3}=10\pi$ cm. $A=\frac12 r^2\theta = \frac12(15)(15)\frac{2\pi}{3}=75\pi$ sq cm.

Example 4 (central angle from a perimeter condition). The perimeter of a sector equals half the circumference of the circle; find the central angle in radian.

Perimeter of sector $= r + r + r\theta = r(2+\theta)$. Setting this equal to $\frac12(2\pi r)=\pi r$: $r(2+\theta)=\pi r \Rightarrow 2+\theta=\pi \Rightarrow \theta = (\pi-2)$ radian.

Example 5 (pendulum path length). A pendulum of length $21$ cm swings through $36°$; find the length of the path it traces. …