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(Continuation of Exercise 37$)$(a) Inscribe a regular $n$ -sided polygon inside a circle of radius 1 and compute the area of one of the $n$ congruent triangles formed by drawing radii to the vertices of the polygon.(b) Compute the limit of the area of the inscribed polygon as $n \rightarrow \infty$(c) Repeat the computations in parts (a) and (b) for a circle of radius $r .$

a) $\frac{1}{2} \sin \left(\frac{2 \pi}{n}\right)$b) $\pi$c) $A_{T}=\frac{1}{2} r^{2} \sin \left(\frac{2 \pi}{n}\right)$$A_{p}=\frac{n}{2} r^{2} \sin \left(\frac{2 \pi}{n}\right)$$\lim _{n \rightarrow \infty} A_{p}=\pi r^{2}$

Calculus 1 / AB

Chapter 5

The Definite Integral

Section 1

Estimating with Finite Sums

Integrals

Integration

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Lectures

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So you have Ah, already under in gun meat over triangles. So, yes, I know. Some travel here with the same I go. You something like that. So this would be a seven. So it's a regular Pouliot, Remember? The angle here is the same for all those. Hey, Mongol. I'm there. Assuming that my my drawing doesn't make you your imagination to work too hard. His distance one. The allure of the angles are the same. So you have on these shape on triangle ladies. Well, peacetime one here. And so, my dear. So you did. And times Well, here for these example, an Eagles seven. So we're very good for General. Aimed a gun. So did I. And because trade, it's people to buy um towards it. What is the area there? What? Is the area off all off? Well, these Yeah, What is Cheerio called that these triangles had it together. So for that would we need to say, Well, my theory of a single triangle, and then we'll take life. Bye. So for the record, Ariel, Ditto Leah with the and I'm going to total area. So area is gonna be ableto in times. Got your triangle? No. Uh, well, you look at these triangle. It has done shake like this are right. Triangle. So here varies. Well, he's not up. Just killed, but he's won. And then this angle there is half How awful that angle. It would be okay with me. Yeah, humps So that these descends here. He's gonna be one of them's sign. So sign off their house. The descends here would be, Go sign off the house, sign your house so that Well, cheerio. These triangle here for the just the berg on these doors store. Because what? The area of the strangle can be seen us twice. The area off these triangle. Great angle for your house. Right. So the air off a single triangle people too. Oh, one one here. One there, because is the least of their So one James school Sino, you know, loves times warm. Well Oh, sane, huh? House? This is the area of a single track. Now, from these, we can see that we have. They're our equals to buy over in so that, um, that Syria real food can vehicle too. One squared coming from one from there, one from there. Uh, times sign off. We want to buy over in these house times. One, huh? That was a sign. Oh, two bye over. And hopes so. The baby is too. These two Jews for Will Castle Eso Well, that that will be good to one squared terms. Sign off by or times sign, but or so that's theory off. One train, the total area. It's a good idea. Peace little area would be in. I'm Stan Diesel vehicle too. Prince, we'll sign. Swan Squirt. That's a sign off by. Over in comes signed. Bye. Where uh so? Well, what is the Lee need? That total area? Not till they re depends on man. So we'll be leaving of it all here. Yeah, thank you. Goes toe. So what we have is gonna be sand goldstein, Katie cams or one word and them's course I know. Bye. Over and thumbs sign off by or now, as you see here is goes through shady on. Uh, because this you see, if you needed times, it's you. What is that? So what we can do here is, uh, since the lineage a single defeat, the vertical sign off and go to one. You can write down at us. Very single safely. The off course casino. Hi. For an times the lean. If both of these leave it exist, then this limit physical to the perp blinks. So for one second that's assumed that this name it exists. No. Here you are. Off limits. Yeah, Pam's zero. So what we can right in us dividing by one over. And what these counseled in these form, it has a form that for which one for which we can apply love it. So because us sign of 00 on one of her and proposed to see you, so apply it will be done. So So Whoa, He's gonna be equal. Drum. There you are. Thieves. We told you already. So is there already? Oh, they're both sign your school sign. Bye. Over in. There's they take off that which is bye. Minus by. Over. Yeah, squirt! Right there. The bees. Bees minus one over. Squirt. So those that was canceled minus this console. So dreamy vehicle too. Oh, I forgot. There. Yeah, by there he's gonna go to the lead. Us You goes to extremity. Why cool sino Bye over. I'm well, this goes to zero. So it's part of living in those one. So it's gonna go to buy so that that Hillary is equal. Bye. And so we'll know in the case, I assumed that by some reason you wanted that those triangles, they don't have to leave. So that that was to worry. Perhaps you want to say, Well, they have made our So what would change in this computation is that you have on are there in order because the realties temple would be are them's sign that's been battling on this thing will be our camp school sign. Sort of. The total area will be r squared cause I know by some fight so that they then expect this computation we get on our school there. So we'll really need are you, since Iris a constant you can pull that constant are doing it. So this thing, it would be well, you can just the constant power willing on the competition of living here remains the same. So, uh so in the case of our off our name gone over a deuce are the digital area will be equal to our square school. Sign off. Bye for end times. Sign off fire and dealing. It is gonna go to I I'm sorry. Square, Um, which is the area of so us. One would expect convergence to the area of a circle. Well, yeah, you understand?

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