Areas and perimeters of sectors (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. 

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:

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

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$12​r2θ

Worked Examples 

Question 1

Question 2

Question 3