Length of Circular Arc & Area of a Sector (using radians)
In this lesson, we are going to look at what radians are, and find new formulas for the the Length of a Circular Arc and the Area of a Sector, using radians rather than degrees.
What are radians?
Up until this point, you've mostly measured angles using degrees. Radians are a different way to measure the size of an angle, based on the number $\pi$π.
Why do we have $360^\circ$360° in a circle?
If we take the number of days in a year, $365$365, and round it off to the nearest convenient number, with lots of factors to make it easily divisible, we get $360$360. This helped astronomers, as each day, the sun would move $1^\circ$1° against the fixed stars.
This is why we have $360^\circ$360° in a circle.
Why would we need another way of measuring angles?
Mathematicians realised they needed another way, more closely related with the formula's for circles, to describe the size of angles. It helps us a lot with calculus of trigonometric functions.
Defining radians
Consider a circle with centre, $O$O, and two points on the circumference, $A$A and $B$B. Radians are defined as the ratio of two lengths in the following way:
Radian Measure: Size of $\angle AOB=\frac{\text{arc length AB }}{\text{radius OA }}$∠AOB=arc length AB radius OA
As all circles are similar to one another, this definition gives as the same angle size, regardless of the length of the radius.
If we consider this ratio, and the arc subtended by a revolution ($360^\circ$360°), we get a relationship for the whole circumference of a circle:
| $1$1 revolution | $=$= | $\frac{\text{arc length }}{\text{radius }}$arc length radius |
| $=$= | $\frac{\text{circumference }}{\text{radius }}$circumference radius | |
| $=$= | $\frac{2\pi r}{r}$2πrr | |
| $=$= | $2\pi$2π |
Therefore, $360^\circ=2\pi$360°=2π.
There are three main variations of this conversion which are important to remember:
$360^\circ=2\pi$360°=2π & $180^\circ=\pi$180°=π & $90^\circ=\frac{\pi}{2}$90°=π2
We can use these relationships to convert between degrees and radians:
CONVERTING DEGREES TO RADIANS:
To convert degrees to radians, multiply by $\frac{\pi}{180^\circ}$π180°.
CONVERTING RADIANS TO DEGREES:
To convert radians to degrees, multiply by $\frac{180^\circ}{\pi}$180°π.
Let's also consider One Degree and One Radian:
$1$1 degree $=$=$\frac{\pi}{180^\circ}$π180°$\approx$≈$0.0175$0.0175 radians
and
$1$1 radian $=$=$\frac{180^\circ}{\pi}$180°π$\approx$≈$57^\circ18'$57°18′
Length of Circular Arc (using radians)
As we saw here, when using degrees, we can find the Length of a Circular Arc using the formula:
Arc Length $=$=$\frac{\theta}{360}\times2\pi r$θ360×2πr
We want to change this to a formula using radians. By substituting $360^\circ=2\pi$360°=2π, 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 |
Often, Arc Length is replaced by $l$l, giving us $l=r\theta$l=rθ
Formula for Length of Circular Arc (using radians):
Arc Length $l=r\theta$l=rθ
Area of a Sector (using radians)
As we saw here, when using degrees, we can find the Area of a Sector using the formula:
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 |
Formula for Area of a Sector (using radians):
Area of a Sector $=$=$\frac{1}{2}r^2\theta$12r2θ
Worked Examples
Question 1
The diagram shows a sector of a circle of radius $5$5 units, formed from an angle of size $2.3$2.3 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{5\pi}{6}$5π6 radians.
Find the length of the arc, rounded to two decimal places.
Question 3
The arc of a circle, radius $13$13 cm, subtends an angle of $\theta$θ radians at the centre of the circle, and measures $11.7$11.7 cm in length. Solve for $\theta$θ, the angle subtended at the centre.
Write each line of working as an equation.