13C

Arcs and sectors of circles

Arc length ℓ = rθ, sector area A = ½r²θ, major and minor arcs and sectors, and the area of a segment.

Arc length and sector area

With θ in radians

$$\ell = r\theta \qquad A = \tfrac12 r^2\theta$$

Arc, sector and segment

The sector is between the two blue radii, measured anticlockwise from the positive x-axis.

Try this

Set $\theta = \pi$. Check that the sector area is half the circle.

Example 1●●

A sector has radius $6$ and angle $\dfrac{\pi}{3}$. Find the arc length, the sector area and the area of the minor segment.

$\ell = 6 \times \dfrac{\pi}{3} = 2\pi \approx 6.28$

$A = \tfrac12 \times 36 \times \dfrac{\pi}{3} = 6\pi \approx 18.85$

Segment $= \tfrac12 r^2(\theta - \sin\theta) = 18\left(\dfrac{\pi}{3} - \dfrac{\sqrt3}{2}\right) = 6\pi - 9\sqrt3 \approx 3.26$

Segment = sector − triangle, and the triangle has area ½r² sin θ.

Major and minor arcs

Two radii split a circle into a minor sector (angle $\theta < \pi$) and a major sector (angle $2\pi - \theta$). Use the matching angle in the formulas.

Segment area

$$A = \tfrac12 r^2(\theta - \sin\theta)$$

Flashcards

Arc length

$\ell = r\theta$

Sector area

$A = \tfrac12 r^2\theta$

Segment area

$\tfrac12 r^2(\theta - \sin\theta)$

Major sector angle

$2\pi - \theta$

Practice

Find the arc length when $r = 5$ and $\theta = 2$.
Find the sector area when $r = 4$ and $\theta = 1.5$.
Find the exact arc length when $r = 6$ and $\theta = \dfrac{\pi}{3}$ (type like 2pi).
Find the area of the minor segment when $r = 6$ and $\theta = \dfrac{\pi}{3}$, to 2 decimal places.
An arc of length $12$ cm is on a circle of radius $8$ cm. Find the angle in radians.
Two radii of a circle of radius $3$ make a right angle. Find the length of the major arc, to 2 decimal places.
Find the perimeter of a sector with $r = 5$ and $\theta = 1.2$.
If the angle of a sector is doubled (same radius), its area:
A sector of radius $4$ has area $24$. Find its angle.