7.13 Arcs and sectors
Length of circular arc (using radians)
Previously we used to find the length of a circular arc by finding the fraction of the circumference of a circle with radius $r$r:
Arc Length $=$=$\frac{\theta}{360}\times2\pi r$θ360×2πr
The formula for arc length using radians is found by considering the definition of a radian, which is the ratio of the arc length to the radius. That is:
| $\theta$θ | $=$= | $\frac{l}{r}$lr |
Rearranging, we get arc length $l$l as:
| $l$l | $=$= | $r\theta$rθ |
Alternatively, if we start with the formula above for arc length using degrees, we can substitute $360^\circ=2\pi$360°=2π, and we get:
| Arc Length | $=$= | $\frac{\theta}{360}\times2\pi r$θ360×2πr |
| $=$= | $\frac{\theta}{2\pi}\times2\pi r$θ2π×2πr | |
| $=$= | $\frac{2\pi r\theta}{2\pi}$2πrθ2π | |
| $=$= | $r\theta$rθ |
Arc Length $l=r\theta$l=rθ
where $l$l = arc length
where $r$r = radius
$\theta$θ = angle subtended at the centre (in radians)
Practice questions
Question 1
The diagram shows a sector of a circle of radius $8$8 units, formed from an angle of size $2.45$2.45 radians.
Find the exact length of the arc.
Question 2
A sector of a circle of radius $4$4 cm is formed from an angle of size $\frac{3\pi}{4}$3π4 radians. Find the length of the arc.
Round your answer to two decimal places.
Question 3
The arc of a circle, radius $15$15 cm, subtends an angle of $\theta$θ radians at the centre of the circle, and measures $16.5$16.5 cm in length. Solve for $\theta$θ, the angle subtended at the centre.
Question 4
A sector of a circle, radius $95$95mm, has an arc length of $159$159mm.
Find the angle $\theta$θ formed at the centre of the circle in radians, correct to two decimal places.
How big is the angle $\theta$θ in degrees? Give your answer to the nearest degree.
Area of a sector (using radians)
Similarly when using degrees, we used to find the area of a sector finding a fraction of the area of the whole circle with radius $r$r:
Area of a Sector $=$=$\frac{\theta}{360}\times\pi r^2$θ360×πr2
We want to change this to a formula using radians. By substituting $360^\circ=2\pi$360°=2π, we get:
| Area of a Sector | $=$= | $\frac{\theta}{360}\times\pi r^2$θ360×πr2 |
| $=$= | $\frac{\theta}{2\pi}\times\pi r^2$θ2π×πr2 | |
| $=$= | $\frac{\pi r^2\theta}{2\pi}$πr2θ2π | |
| $=$= | $\frac{1}{2}r^2\theta$12r2θ |
Area of a Sector $=$=$\frac{1}{2}r^2\theta$12r2θ
where $r$r = radius
$\theta$θ= angle subtended at the centre (in radians)
The area of a triangle in a sector
Some questions may require you to find the area of the triangle in the sector formed by a chord joining the ends of the arc. In this situation, we would need to use the area of a non right-angled triangle formula: $A=$A= $\frac{1}{2}ab\sin C$12absinC
Major sector or arc = sector or arc subtended by an angle $>180^\circ$>180°
Minor sector or arc = sector or arc subtended by angle $<180^\circ$<180°
Both formulas require the angle to be in radians so if a question gives you an angle in degrees, you need to convert it to radians first.
Practice questions
Question 5
Find the area of a sector of a circle with a radius of $24.1$24.1 meters and a central angle of $\frac{4\pi}{5}$4π5 radians.
Give your answer correct to two decimal places.
Question 6
Find the area of the following sector of a circle in square centimetres.
Give your answer as an exact value in terms of $\pi$π.
Question 7
Find the area of the following segment of a circle in square centimetres.
Give your answer as an exact value in terms of $\pi$π.
Finding the radius and angle of a sector given its area
If we are given, the area and the radius, or, the area and the measure of the central angle, we can calculate the missing information by rearranging the area formula.
Worked example
Example 1
The sector below has an area of $8$8 m2 and a radius of $2$2 m. Find the value of $\theta$θ in the sector below.
Think: We are given the area and the radius of the sector so we can use the formula $\theta=\frac{2A}{r^2}$θ=2Ar2.
Do: Substitute the values into the equation:
| $\theta$θ | $=$= | $\frac{2A}{r^2}$2Ar2 | (Writing down the equation) |
| $=$= | $\frac{2\times4}{2^2}$2×422 | (Substituting) | |
| $=$= | $\frac{8}{4}$84 | (Simplifying the top and bottom) | |
| $=$= | $2$2 radians | (Simplifying the fraction) |
Practice questions
Question 8
The sector below has an area of $3.4$3.4 cm2 and a radius of $2$2 cm.
Find the value of $\theta$θ in radians.
Question 9
The area $A$A of the minor sector $OAC$OAC shown below is $252\pi$252π cm2. The original circle had a centre at $O$O and a radius $r$r measuring $42$42 cm.
Find the value of $\theta$θ, the measure of the angle $\angle AOC$∠AOC.
Give your answer in radians in terms of $\pi$π.
Find the exact length of the arc $AC$AC.
Give your answer in terms of $\pi$π.