a-let-a_n-be-the-area-of-a-polygon-with-n-equal-sides-inscribed-in-a-circle-with-radius-r-by-divid-3

Question

(a) Let $A_{n}$ be the area of a polygon with n equal sides inscribed in a circle with radius $r .$ By dividing the polygon into n congruent triangles with central angle $2 \pi / n,$ show that
$$\mathrm{A}_{\mathrm{n}}=\frac{1}{2} \mathrm{nr}^{2} \sin \left(\frac{2 \pi}{\mathrm{n}}\right)$$
(b) Show that lim $_{\mathrm{n} \rightarrow \infty} \mathrm{A}_{\mathrm{n}}=\pi \mathrm{r}^{2} .[$ Hint: Use Equation 3.3 $.2 .]$

Ayman Sherif · Accepted Answer

Okay. Hi. And this problem, we have a circulates Inside the circle is appointment of equal size, which can be by this. This is a center of the circle. This parliament is divided into a number of equal triangles. For example, this is this is one of this is one thing This is the next is the second language. This is the same triangle and support string this is this site is or And this site is also which is a sent a radius of the second. This angle is the central anger which is off. If you want to get the area off this triangle, we&#x27;ll be acquitted. Video, I hope deploy. But Roy Oh, off deploy swing is equal to boy over. And where n is the total number of triangles in sizable in this case with before so But we want to get the area of the overall volume. So you way will not deploy this area by end, which is a number off trains it so in which, which is a total area of the Taliban with the equal or squared Uh, sign Do boy deploy, which is the first requirement, is the second requirement. So we we knew that the area of the bargain will be equal and it will deploy or square deployed and oy, sign to over in. We want to take limit to this equation so limit from then turns to infinity. Half are square is a constant eyes is very circus constant So it will be equal hope squared limit when tends to infinity Oh, look it sighing to over in We will take this equation and multiply it mainly by one. But in our case it will be too boy over do for So this limit will become I will show in exploit to will deploy and over to boy must deploy sewing Cool boy boy over in if we talk If we take this turn on and make it a diminished denominator every year it would become equals to guard no deploy toe over and over to boy over in Let system be seated which is are some thrilling So in the next flight limit when end girls 24 in will be equal Oh squared deploy limit when sita zero Because what we&#x27;re increasing the number and the sita or the central angle will decrease gradually until it nearly become zero Flemington. See? It tends to zero toe boy Want deploy saying see you seek mint. We will take the two boy outside so exploit limit and well in turns to infinity will be acquitted. Pool off, squared, deploying to boy Look, Chloe, limit tens zero So I&#x27;m so in Saint Overseas. The tour Big gun with the two. So it will be equal. Sister is one. This time would be equipped to one because we see the term is 20 signs. It was one through This time is one. So this term will be equal next light limit and then turns to infinity. Boy R squared one. She could Boy. Oh, squish. This is a quote on This is the overall area of the city. This is the way as they used toe proof as overall area of the second as shown, uh, isn&#x27;t it you? Thank you.

Frank Lin · Answer

So here we use untrue I and goes to approximate a polygon to approximate area of the circle. So here, so it should eyes try. Go on, look at this. This one has radius. Are this wise radius? Are this is one over the the horse Sir Ho sang Go So the whole circle goes to part over Wei have learned from the high school that the area of the triangle this one over two length of this one times Lance off this one time sigh of this and go Oh, so it&#x27;s not yet also distressed area of the triangle is one over two times dance of this side and lands of this side Time side off the angle to pi over and the alien is simply trying goes at it together Therefore he gives us up This formula and computer are limit on fraud on course the infinity here that is this so here we still have to take limit here We put on over to pie here that we put pie are square here and we have side you pie over this just below there How to brought shake the limit here on this term This part it is. Actually, we actually use a famous result from calculus. What unconsciously infinity? This argument this too high over and goes to zero. And this can be view us divided by two pi over. So we used the famous result that science over hacks the limit of ab support zero give us one So this part of the limit will goes to one and we&#x27;re left with pi r squared.

Carlos Pinilla · Answer

to have the following problem. There&#x27;s Ah, there&#x27;s some pizza for you want to slice? I want to be fighting D&#x27;oh! And he calls license. Oh, so he&#x27;s not perfect drying, but you already have. It wasn&#x27;t that bad angle where the same for all of these pieces. So I go there way have some radius here. So that way we do the whole turn you get, Uh, Bye on then. Each of these lice is us. Mango feta on olders and of them and his laces. So you want to figure out you have those in his license and the radios? What is the area? My theory of that shape. You there? Well, uh, if you believe that his life was the same Oh, area. Uh, What&#x27;s gonna be in times? It&#x27;s a van. Where s event is the area of a single slice here? Not shaded area there. That&#x27;s one of those pieces. A seven. So for his event, you have been following setting. So they&#x27;re triangle. Do you have the sango? Huh? On the his distance, there is the same LaSalle. This knows there are all right. So so loose travel would Well, we can cut it in how unfolded to see that the dollar here, it&#x27;s gonna be ableto Well, if you&#x27;re full Did is gonna vehicle to the area. Piece tangle. Right. We&#x27;re going for here. You have this angle. That&#x27;s Hobbs. That is the same wood there hops. Well, if you do, we have these triangle one of those. Well, these your triangle is the same. Was that about over travel? So here we are. No. Perhaps you have are. But our here&#x27;s be sense. So well, the area of these rectangle I was gonna be these distance. These those picks legal Does the snow&#x27;s X here is that the snow is white. All right, extends. Why, that is the area office when eggs, That&#x27;s why. But what is what is sex exists? Are was signed off the house. Our cause. Books? No, that is Baron. That is why are called on. Uh, that is way on eggs. Yeah. My butt on X is equal to our sign. Perhaps sign of her house. So, uh, also that if he did a product, you get the delivery is gonna be our squirt. We&#x27;ll sign off their hooks, then sign off the rocks. Sign off in the house on the DL angle. Identities tell you that these gadgets, they&#x27;re very simple Toe. What? Huh? Off sign off the that I&#x27;m going in their place. So one how both sign will be Sango two times two attacks. So that was very nice thing happens here. As you can guess, the two goes away, but and so you have area will still have the arts. Quarter square little area of these single piece will be of a single&#x27;s life. It&#x27;s gonna be our square loves sandal, but, uh Well, uh, what is it? Well, we have in his laces. So if there should be such that and that was that I go gives you the whole turn Did you buy? So we can be fired, but and wonderful. And they get this cancels so later we got to find a way to buy him. Great. So this area is abandoned on a vehicle toe our square replacing for what it is. Sign off to buy over in. So as we if it is drawing, we see that you Syria is cause trying to leave. Oh, it&#x27;s not. It can be greater than here of the circle because he&#x27;s constrained there so he can ask yourselves little area there. This deal is that how easily the same goes for you. Katie is early. Need us saying go to infinity. Hey. Hey, Here is that little area. So sometimes it&#x27;s a band is gonna be ableto in terms. This so is their lead to that area, Which will? There should be vehicles. You got these constrained to stay within the circle. So what is the lead off? N times are square hubs square house. So, you know, bye over in. And if you just go ahead the Bailey quitting Cy Baby, will you get the beauty times are square house That was signed off. All the results to zero. This is zero, but this is zero. So serious times infinity. What is that? But I will when you lose, you can manipulate this and already does, uh, wanna burn so peace Can we read us? Her square huffs sino good bye over end. We can bring that What? Multiplying white and is the same musty by biting by Want over so that well you think you are I&#x27;m going Divinity. We&#x27;re gonna have a local form, so it goes to zero zero or zero. Yeah, we&#x27;re gonna blow people. So after a fine no pickle Hello? Get her in. Shape it up with respect to end. So you&#x27;re The sign is that is what we have been through. They are square house people Sign is oh, sign, but well, the sign of the same But we have your depression with respect And so you get a factor of minus two by over a squared over since word And then we differentiate the bottom with respect when minus one Over in squared I said I&#x27;d Well, something wonderful happens here. I wonder where it ends where it comes to that so that well, you clean equals angle into beauty. You&#x27;re gonna have to co sign off zero, because by wearing goes baby Mr Zero. So, cause I hope, Cyril, but he&#x27;s gone. Okay, so So these whole expression is gonna go to grows through. Uh, so are square Hobbs. Them school sign of 01 times one times to buy them. Still buy. I do. But until another nice thing happens here to to to get that the leader is the area. Oh, the circle are scored in spite, but he&#x27;s here, not circle. Cool

Fuzail Shakir · Answer

Okay, So if we have any equal scientific radius are we have to show that the area is equal to 1/2 and our square times sign off to pi over and so this fairly easy to show. So let&#x27;s suppose we have a triangle with three sides. Okay, so therefore, the area of the triangle with three sides has the equal 1/2 1/2 times, three times the really square times sign off to high over three. Simply because if you said this is equal to the radius and then the sine of the angle is thehe mount of the circle that is being covered so we could set this The amount of circle that it&#x27;s being covered and we have pie are square to be the area of the whole circle. So you basically can get an area of the amount of the circle there is being covered multiplying by. Of course, you have to cut it off because if this is a circle, you&#x27;re not counting the red part. So therefore this theory must be approved

Chris Smith · Answer

problem. Number 22 went ahead and solved this out Just kind of conceptual problem. And I wanted to go ahead and have so doubt before stumbled all over it in the video we were asked to do here is inscribed inside of polygon, which I have done in red here, inside a circle of radius. One report a, um and then compute the area of one of the integrate triangles formed by drawing radio, which is here in blue from they&#x27;re in the circle out too, to the each. Vertex. Yeah, makes all these triangles in the green shade. Here, here is the area that we&#x27;re trying to calculates. I did a five sided polygon here, but we&#x27;re gonna keep said they used in five who used in so that we can keep this in terms of an end sided polygon. Not just a five sided where height cider, No 20 sided polygon. Whatever in you want to use comply that in there. So to find this area of the green of each of these triangles, what can we do? We know that the area of a triangle is 1/2 times the base times height. So the height tears from the tip of the triangle and two perpendicular to the base. The base is this whole link. And so how can we find those two dimensions being eight? We know that there&#x27;s 360 degrees inside of a circle. And if we have an inside a polygon, that means you have in triangles each trying August 360 divided by in of the 360 degrees. And so this angle data here that have marked yeah is actually only half of that. So divide that by two get data is 180 degrees divided about in. And so then we have a rock star here. You can see and we confined decides the legs of these right triangles by using a little bit of trig. And I made this other dimension here base of one that&#x27;s actually gonna be half of the. So the sign of data is gonna be opposite side over the high part news be won over are so be one is our Cynthia, And I&#x27;m actually going ahead and doing this part. See first, because if we can do part, see you just plug in one for our news at the party. So and then the Costa is the Jason side over that partners. So I get h over our and so h is our Times coast. And so this big dimension be is actually a double of beasts of one. So we could be is to Argentina in ages. Our coast aren&#x27;t there. So the area of the triangle which party is asking us for the area of one trying Not all the triangles, but just one. There&#x27;s 1/2 base times height, so 1/2 times be be times eight and then the twos cancel. So the area ISS is our sine data, which is 180 over in Look back up here and our coast Santa, 180 over hand. And then you don&#x27;t find that which are squared. And that&#x27;s the area one triangle that&#x27;s made by inside a polygon and drawing on the radio. And so, for part A, we won&#x27;t when the radius is of the circles Once will be bugging one here, one squared and we just get rid of the r squared. Got sign of 1 80 over end times co sign of one any over him, and that&#x27;s all. We got a report now for part B. We want to compute the limit of the area of inscribed polygon, as in approaches infinity. So if the area of one triangle is is given by here in part a. Well, we have an inside of polygon. There&#x27;s in triangle, so we just multiply by 10 times what we already had for part A. And that&#x27;ll be the area of the whole polygon, not just area of one single trying. And if we take the limit of this is in approaches infinity. You know that this and it&#x27;s gonna keep going, going. It&#x27;s going approach Infinity. This sign of 1 80 over and the 1 80 over and is in gets really big. It&#x27;s gonna get a zero for the sound&#x27;s going zero. So how do we deal with that and not so mathematical scientific terms? This and it&#x27;s getting really big, and it&#x27;s getting multiplied by very, very small number. And actually, what happens when you won&#x27;t buy those together in is getting very large. It actually becomes Pa. If you don&#x27;t believe me, get out your calculator and try 100 times the sign of one of the over 100 try 1000 times kind of winning me over 1000 and then do a 1,000,000 times time leg over me and see if you don&#x27;t get closer and closer pie. So yes, and goes to infinity end times decided waiter, and actually becomes pot and coast on is just warrants of the area of the polygon becomes, huh? His in goes to infinity and, uh, interesting enough our circle with raise One area of a circle is pi r squared If our is one just equals power and that&#x27;s what we get here. So we have an infinite number of sides for a polygon. It just becomes the area of a circle and go back to part. See, we want to do the same thing. Like that limit is in approaches. Infinity with a circle of the generic radius are so those that are squared back in there. We get the same idea because we still got this end time sign of 1 80 over again. And is that ghost? They&#x27;re family. That multiplication is gonna become pie. This coast on terms gonna go one. So, uh, times R squared. So the area of the polygon is Yoon approaches. Infinity is power squared, which is the radius of a circle. And that&#x27;s what we expected

Brittany Knowlton · Answer

in case this is another three part question. So it says that we have this inside polygon inside a circle of radius one, and we want to compete. The area of one of the N congruent triangles first get individual. So we have this circle with a radius one, and we&#x27;re gonna have all of these triangles apologize from picture and keeps going. And there&#x27;s a me end of these. So burn and just look at one of these triangles here. I&#x27;m gonna pull it out here just so we could have a closer look at it. No. Okay, let&#x27;s say that&#x27;s this one are here. So we have this, I softly strangle we known both of these. Sir Radius is going to be one, so we can actually just split this into two triangles too. Right? Triangles? Where&#x27;s my pot? News is going to be equal to one, and then we can find the base in the heights. But if you remember, you&#x27;re so Catella Rules. So we have So Kito asshole sign gonna be opposite over iPod news and so then are outside to this one. It&#x27;s just gonna be our pot News times The sign of this angle, which we know from before that that acute angle is just going to equal to pi over two times and cause and is the sides of our polygon. So it&#x27;s just gonna equal pie over and and we&#x27;re in our ages, just one. So now we know that our right this sign of pie over and our base, it&#x27;s co sign of pie over end. And so now we know that the area of our triangle up here, it&#x27;s just gonna be We have to the zone and put it to you out here since we&#x27;ve again month divide that triangle too, right, triangles. And then we really know that the former and left for your triangle is one half times base times height. So no one half here my height again. Is that sign of pie over in. And then times co sign a pie over in which, if you remember, your trig identities sign Chief data is equal. Teo to sign if they tako science data. Well, we see that we have out here, so this can be simplified, Tio, what we have here. So one half sign we&#x27;re doing Tuesday buses going to sign of to pop over. And so this is gonna be the answer to the first part of that question. Okay, Snow for part B, we want to compute the limit of the area of the inscribe polygon as an approaches infinity. So before you found area, just triangle. Now we&#x27;re going to find of the whole pa Leon. So that&#x27;s just going to be n times the area, the triangle since we have in triangles. So that&#x27;s just gonna equal and over, too inside of two pi over. And it&#x27;s not gonna take the limit as an approaches infinity of that. So I know this If I plug in and his poaching Fendi right now here I end up with infinity here. If I put in and hear this whole thing&#x27;s been go, zero sign of zero goes a zero saiget times here here, which is an indeterminate form. So we can&#x27;t have that. So we&#x27;re gonna have to manipulate this equation. I see how the sine of to pry over end right here and if you remember also from you&#x27;re right tricks with limits, the limit as Expro jizz Mindy, A sign of X over X is just equal to once I want to try and create that on this side someone and go ahead and multiply this by what I have inside signed right now, so to pry over end over t pire end. So when I do that when I did two prior end times this I see my end cancels out, my two cancels out some just left with the pie. And then I have exactly what I wanted over here. So sign of Sioux pyre and divided by two pie or n So then I take that limit, this whole thing, it&#x27;s just going to go to one side. Then I just end up with pie. Then for the last question is asking us to repeat the computations in part and b for a circle of radius are. So if we notice over here, that&#x27;s just gonna change. Where are partners is, so this would have been our instead. And since I have that for both of sign and co sign for my base in height, that just means I&#x27;m going to be multiplying each area by r squared so that in my area of the triangle, then become one half r squared times. Sign of two pi over end in the area of the polygon. Well, then be exactly that. T just multiplying by r squared. And when I take the limit, it&#x27;s an approaches infinity of that using the same trick just before we&#x27;re going to find that that&#x27;s just going to be pi r squared, which, if you notice that is exactly the area of the circle.

Problem 26 Medium Difficulty

Answer

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

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

Calculus Early Transcendentals

Catherine R.

Anna Marie V.

Kayleah T.

Michael J.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartCalculus Early Transcendentals

Catherine R.

Anna Marie V.

Kayleah T.

Michael J.

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

James Stewart

Calculus Early Transcendentals