 # In trigonometry, the area of a sector of a circle is the segment of the circle. Its area is equal to half of the product of the radius and its arc length.

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Hey, guys, we're gonna talk about some examples of how to find the area sector in any question that's kind of related to the area of sector. Let's suppose we have this circle. We know that this angle here is 240 degrees and that the radius of the circle is 13. The question is, can you find the area of the shaded region aka Can you find the area? That sector so recalls all about proportions. You could say to 40 is to 3 60 as the area of our sector is to remember, the area of the circle is gonna be pi times 13 squared, which is gonna be 1 69 pie. So if I cross, multiply or just multiply by 1 69 pi, I get 1 69 pi times to 40/3 60. That's gonna equal Alfa So 1 69 times 2 40 divided by 360. That's gonna get us 3 38 3rd pi Or, in other words, we're gonna get something roughly in the ballpark of 353 95 square units equals Elva, which be the area of your sector, and it makes sense, guys. 13 squared pie. It's 1 69 pi. The area of the whole circle is 530.9. So it makes sense that, you know, this being more than half of the circle, the whole area being 531 roughly. And we're getting an area of 354 roughly that completely fits with my estimation. Feels great. Okay, What if I have a circle? Okay. And let's say I know this angle here. It's 70 degrees and let's say I know that the area of the sector is let's say, 20 square units. What? Could I ask myself? What I could say? Hey, what's the radius of the circle? What is this equal? Well, I could say, Well, I know the 70 is 23 60 as 20 is to the area of the circle, which is pie R squared. Okay, so what I can do here is several things I can cross multiply. It's gonna get me 70 times. Pi r squared equals 20 times 3. 60. Clean this up a little bit 20 times 3 60 equal 7200. So then we have 70 pie times are squared equals 7200. All I have to do now is divided by 70 pie. So 7200 divided by 70 high gets me roughly 32.7. That's equal to R squared. And I should change this to roughly. And then I'm gonna square root both sides. Yeah, and I'm gonna get are to be roughly 5.72 units. So here are two examples of how you can move forwards or backwards, depending on what you're given. The key is the proportionality angle to the entire revolution of 3. 60 equals either the arc length or the area of the sector to the circumference or the area of the circle, respectively. Okay, I'll see you next time. Thanks. Webster University
##### Top Geometry Educators ##### Catherine R.

Missouri State University  ##### Kristen K.

University of Michigan - Ann Arbor ##### Samuel H.

University of Nottingham