continuation-of-exercise-21-a-inscribe-a-regular-n-sided-polygon-inside-a-circle-of-radius-1-and-c-2

Question

(Continuation of Exercise $21 . )$
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 $r .$

Brittany Knowlton · Accepted 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.

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

Carlos Pinilla · Answer

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&#x27;s a regular Pouliot, Remember? The angle here is the same for all those. Hey, Mongol. I&#x27;m there. Assuming that my my drawing doesn&#x27;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&#x27;re very good for General. Aimed a gun. So did I. And because trade, it&#x27;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&#x27;ll take life. Bye. So for the record, Ariel, Ditto Leah with the and I&#x27;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&#x27;s not up. Just killed, but he&#x27;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&#x27;s gonna be one of them&#x27;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&#x27;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&#x27;s theory off. One train, the total area. It&#x27;s a good idea. Peace little area would be in. I&#x27;m Stan Diesel vehicle too. Prince, we&#x27;ll sign. Swan Squirt. That&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s assumed that this name it exists. No. Here you are. Off limits. Yeah, Pam&#x27;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&#x27;s gonna be equal. Drum. There you are. Thieves. We told you already. So is there already? Oh, they&#x27;re both sign your school sign. Bye. Over in. There&#x27;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&#x27;s gonna go to the lead. Us You goes to extremity. Why cool sino Bye over. I&#x27;m well, this goes to zero. So it&#x27;s part of living in those one. So it&#x27;s gonna go to buy so that that Hillary is equal. Bye. And so we&#x27;ll know in the case, I assumed that by some reason you wanted that those triangles, they don&#x27;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&#x27;s sign that&#x27;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&#x27;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&#x27;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?

Ayman Sherif · 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.

Chris Smith · Answer

improbable Number 21 were asked to inscribe regular inside a polygon inside a circle of radius one and compute the area of the polygon for the following values of end support. A we wanted Thio. Where are four sided polygon? So I have a circle with a raise of one and we want to inscribe a square which would be yes on. I&#x27;m not gonna sit here and calculate the areas you can go. Looked it up on your own. That&#x27;s not the point of this problem so much. Waste your time with area calculations here and showing working all that out. Basically, you can use this and find them links. You need ejaculate area of the square So the area here of the square is going to be respect. Red area square, four sided polygon It&#x27;s going to be zero 0.707 units squaring say feet square, major squared Whatever. Whatever The same years if if the radius is one meters be points at most seven major squared part B, we will use an eight sided Oh, you know our octagon. So I have a sorry about the little breaking video there. Ah, little interruption. But anyways, Part B is a were inscribing an octagon. So we got eight sided polygon are one too three or 78 Something like that. You have a radius of one. And so our area here eight sided polygon is again. You can go look up how to do this. I&#x27;m not gonna waste your time with that. Our area is 3.6 one units squared. Whatever. Very Yitzhar, are you in court? See, we&#x27;re using a 16 sided polygon. Um, whatever. That&#x27;s gonna look like one, Thio 3456789 10. Well, 30 40 50 60. Sorry about that, but I don&#x27;t think you get the idea of what we&#x27;re doing here. 16 sided polygon regulars bottom in all the side length of the same. And all the angles are saying so are 16 sided polygon inscribed in a circle radius, one that&#x27;s gonna have an area of two no more. Oh, my apologies. Um, have actually switch this around. So this this is actually had these calculated out before, And this is extra trip No. 61 and the eight sided pulling on ISS 2.8 to 8. Oh, sorry about that, um, little mistake there, but so what&#x27;s happening here in the crux of this problem that I really need to understand is that if you notice the more side you get here, the more the closer you get to the race in a circle, um, Mrs Party, that the area of the circle is just power squared the rays of Warren. That&#x27;s pie, which is 3.14159 So if you know this, the more signs we add we get closer and closer to the true at value for the area. So and that&#x27;s kind of like when we&#x27;re looking at these curves and the area under graphs. Well, what do you use? Rectangles will use these rectangles, and we used great big rectangles like this. Then you&#x27;re not too close. But if you go on, use a bunch of little bitty rectangles, your area approximation gets better and better. So that&#x27;s what the idea of this problem is, is that the warning does get as close to the area of the circle as we can if we&#x27;re approximating that using polygons, the Maur Paula got the more sized apology on the closer the heir to the area circle that we get

Problem

In Exercises $23-26,$ use a CAS to perform the fo…

Problem 22 Medium Difficulty

Answer

a) $\frac{1}{2} \sin \frac{2 \pi}{n}$
b) $\pi$
c) $\frac{1}{2} r^{2} \sin \frac{2 \pi}{n}$ and $\pi r^{2}$

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a) $\frac{1}{2} \sin \frac{2 \pi}{n}$b) $\pi$c) $\frac{1}{2} r^{2} \sin \frac{2 \pi}{n}$ and $\pi r^{2}$

Thomas Calculus

Anna Marie V.

Heather Z.

Kayleah T.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

George B. Thomas Jr.Thomas Calculus

Anna Marie V.

Heather Z.

Kayleah T.

Samuel H.

a) $\frac{1}{2} \sin \frac{2 \pi}{n}$
b) $\pi$
c) $\frac{1}{2} r^{2} \sin \frac{2 \pi}{n}$ and $\pi r^{2}$

George B. Thomas Jr.

Thomas Calculus