Problem

Use a CAS to perform the following steps. a. Plot…

04:37

Problem 22 Easy Difficulty

Continuation of Exercise $21 .$ )
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.
Compute the limit of the area of the inscribed polygon as $n \rightarrow \infty$
- Repeat the computations in parts (a) and (b) for a circle of radius $r$

Answer

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Calculus 1 / AB

University Calculus: Early Transcendentals

Chapter 5

Integrals

Section 1

Area and Estimating with Finite Sums

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Video Transcript

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'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're trying to calculates. I did a five sided polygon here, but we'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'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'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'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'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'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't find that which are squared. And that's the area one triangle that's made by inside a polygon and drawing on the radio. And so, for part A, we won'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'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's in triangle, so we just multiply by 10 times what we already had for part A. And that'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's gonna keep going, going. It's going approach Infinity. This sign of 1 80 over and the 1 80 over and is in gets really big. It's gonna get a zero for the sound's going zero. So how do we deal with that and not so mathematical scientific terms? This and it's getting really big, and it's getting multiplied by very, very small number. And actually, what happens when you won't buy those together in is getting very large. It actually becomes Pa. If you don'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'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'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'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's what we expected

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