Area of a Sector

Derive the formula for the area of a sector.

Finding the area of a sector is exactly the same as finding the length of an arc.

Here are the key things to remember whenever you're doing these problems

• • An angle is a part of a full rotation

    • a full rotation would be 360°

  • A sector is a part of the area of a full circle

    • the full area of a circle is A = π r²

Since an angle is only a fraction of a rotation, we can only use that fraction of the area. Let's look at an example.

Here we have a sector L with subtending angle 30° and a radius of 24 inches. We want to know the area of L. First let's find the area of the whole circle, just like in 7th grade.

A = π r²

= π x 24 in x 24 in

= 576π in²

But L isn't the whole area, it's only a fraction of it. So now we need to figure out what that fraction is.

To find the fraction, we need to look at our angle. We ask, what fraction is 30° of a whole circle (360°)? Remember, a fraction means part over whole

For our angles, our part is 30° and the whole circle would be 360°. So the fraction of circle we're working with is

= =

So we're only working with 1/12 of a circle.

Well now that we know that, we know we must also use 1/12 of the area. Since we already calculated that our area is 576π in², we just need to multiply it by 1/12

L = x A

L = x 576π in²

L = 48π in²

This is a great way to think about it, and I recommend you calculate sectors in this style. However, eventually, you may want to have a formula to do everything for you. We know that a sector is a fraction of the full area of a circle, so let's start here

L = x A

We found that it's easy to find this part over whole by using the angle.