- Let's go ahead and take a look at the area of a sector. A sector of a circle is the portion of the interior of a circle intersected by a central angle. So here we have a sector that is colored green.

You can think of this as a piece of pie. If the central angle for a sector is measured in radians, then theta divided by 2 pi makes up the fraction of a complete circle with area pi r squared. Therefore, the area of a sector is theta divided by 2 pi times the area of a circle. And here we can see the pis will simplify out, and we're left with 1/2 r squared theta, where theta, again, must be in radians.

Let's take a look at an example of this. Crops are often grown using a technique called center pivot irrigation that results in circular shaped fields, as you've probably seen from an airplane. If the irrigation pipe is 450 meters in length, what is the area that can be irrigated after a rotation of 240 degrees?

So the irrigation pipe of 450 meters would be this length here. And so if it rotates 240 degrees, it would carve out this area in green or water this much land. So let's go ahead and figure out what the area of this land would be.

So our area will be 1/2 times 450 meters squared times theta, which we'll have to convert to radians. We have a common factor of 60 between these two. There are three 60's in 180 and four in 240.

So we have 4 pi divided by 3 radians. Let's go ahead and go to our calculator. Let's first find the exact value, and then we'll find a decimal approximation.

So to find the exact value, we'll leave off the pi until we get to our final answer. So we have 1/2, or 0.5, times 450 squared times 4/3. Again, we'll add the pi back in our final answer.

So we have 135,000 pi square meters. Again, we're dealing with area now. So we have square units. And let's go ahead and find the decimal approximation here by multiplying by pi. So we'll just take this previous answer and multiply by pi. And we have approximately 424,115 square meters.

OK, I hope you found this video helpful. Thank you for watching, and have a good day.

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