Area and arc length formulas of a sector when the angle given is in radians.


The formula used for working out the arc length of a sector when the angle is given in radians is:

l = rѲ

Similarly, the formula for working out the area of a sector when the angle is given in radians is:

A = 1/2r²Ѳ

l is the arc length, r is the radius, Ѳ is the angle inside the sector and A is the area of the sector.

These formulas are much easier to use than the formulas used for sectors when the angle is given in degrees as the formulas in radians are much shorter.

Let’s take a look at some examples at working out the arc length and area of a sector when the angle is given in radians:

Example 1

Work out the arc length and area of the sector shown above. Leave your answers in terms of Pi (exact solution)

Here, Ѳ = 4π/15 rads and r = 10m

Let’s work out the arc length first:

l = rѲ

= 10 × 4π/15

= 40π/15

= 8π/3 m

Next work out the area of the sector:

A = 1/2r²Ѳ

= ½ × 10² × 4π/15

= 50 × 4π/15

= 200π/15

= 40π/3 m²

So the arc length of this sector is 8π/3 m and the area of the sector is 40π/3 m².

Example 2

A sector has an arc length of π/3 cm and a radius of 5cm. Calculate the area of the sector. Like example 1, give your answer in term of Pi.

Before, you can work out the area of the sector you need to work out Ѳ.

You can find the angle by rearranging the arc length formula:

l = rѲ

rѲ = l (divide both sides by r)

Ѳ = l/r

Now sub in l = π/3 and r = 5 cm:

Ѳ = l/r

= (π/3) ÷ 5

= π/15 rads.

Now since you have the angle inside the sector then you can now use A = 1/2r²Ѳ to work out the area of the sector:

A = 1/2r²Ѳ

= ½ × 5² × π/15

= ½ × 25 × π/15

= ½ ×5π/3

= 5π/6 cm²

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