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  Sectors (Trigonometry)


A sector is a fraction of a circle. Basically, it’s a wedge, or a slice of pizza.

Any problem involving sectors is designed to intimidate, since precalculus classes don’t usually spend time on sectors; of those that do, most spend relatively little time, and for most students, it was a long time ago anyhow.

So the College Board expects you to hit a problem like this and panic, because you assume you’re being asked to remember something you have practically no chance of remembering.

The good news is, there’s really hardly anything to remember. Any problem using sectors requires two things of you:

  1. Figure out what fraction of its parent circle the sector is, and
  2. Figure out whatever property of the circle the question asks about

So, if I have a sector with a 40-degree central angle and a side length (i.e. radius) of 3, and I want to know its area, I just observe that:

  1. 40 degrees is 1/9 of 360, so this wedge is 1/9 of the parent circle
  2. The parent circle has area π(3)2, or 9π

...and therefore this sector has area 1/9 of 9π, which is π.

Similar math shows us that if this sector is actually a pizza slice, then it has 2π/3 units of crust.

Please verify these results to make sure you understand what’s going on here.