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Area of a Sector - Concept
A sector in a circle is the region bound by two radii and the circle. Since it is a fractional part of the circle, the area of any sector is found by multiplying the area of the circle, pi*r^2, by the fraction x/360, where x is the measure of the central angle formed by the two radii. The area of a sector is also used in finding the area of a segment.
Let's say you wanted to find the area of this region right here that I've shaded in, we'd call that a sector. In order to calculate the area let's start off with what if we had the whole circle? Well the area of the whole circle is pi times the radius squared. If we're talking about just one little piece however and I told you that this was a right angle you'd say Mr. McCall that's pretty easy, the area of that sector is going to be a quarter of one fourth of the total area where the total area is pi r squared. However I want to generalize this formula for any type of sector. So if we know that this x degrees and if we know our radius. We're going to say the area of the sector is going to be the area of the overall circle times this fraction and one fourth is actually the simplified version of 90 out of 360. Where the top, where the numerator is your intercepted arch or your central angle and the denominator is how many degrees you have for the whole circle. So I'm going to say this is times x out of 360 where x is the intercepted arch or the central angle. So I'm going to say area of the sector is equal to the area of the whole circle times whatever fraction you're taking off that circle.