geometry-the-area-of-a-regular-n-sided-polygon-inscribed-in-a-circle-of-radius-1-is-given-by-afracn2

Question

GEOMETRY The area of a regular $n$ -sided polygon inscribed in a circle of radius 1 is given by
$$
A=\frac{n}{2} \sin \frac{360^{\circ}}{n}
$$
(A) Find $A$ for $n=8, n=100, n=1,000,$ and $n=10,000$ Compute each to five decimal places.
(B) What number does $A$ seem to approach as $n \rightarrow \infty ?$ (What is the area of a circle with radius $1 ?$ )
(GRAPH CANT COPY)

Peter Winans · Accepted Answer

all right. We have a polygon inscribed in a circle, and we have a formula that calculates the area of that polygon where a is equal 10 divided by two times to sign of 3 63 160 degrees that is divided by N. And we know the radius of our circle is one. So we&#x27;re going to calculate the area of this polygon that&#x27;s inside this circle with a radius of one for several values. We&#x27;re gonna start where N is equal to eight. So the area is going to be eight divided by two times the sign of 360 divided by AIDS. And when we calculate that we get approximately 2.8 to age for and then we do this for N is equal to 100 at 1000 will get the 1000 next. So area there&#x27;s gonna be 100 over to times the sign of 360 over 100 and that&#x27;s going to be approximately 3.13 953 and next we&#x27;re going to do n is equal. 1000. We get a equals 1000 divided by two times the sign of 360 divided by 1000. And when we calculate this out, we should get approximately 3.1 for 157 No, I asked. We&#x27;re going to calculate our area when n equals 10,000 and then is representing. By the way, I didn&#x27;t mention this, but N is representing the number of sides in the polygon. Right? So we&#x27;re increasing the number of sides we started with eight sides on 101,000. Now we want to know what the area of a polygon that has 10,000 sides is if it&#x27;s inscribed in a circle, has a radius of one. Well, our area there&#x27;s gonna be 10,000 divided by two times the sign of 360 divided by 10,000 in this is approximately gonna equal three points. 14159 Now, if we look and all of these values When we had eight sides, we had approximately 2.8. When we had 100 sides, we had approximately 3.13953 When we have 1000 signs, we have approximately 3.14157 10,000 signs. We have approximately 3.14159 So as and is increasing right is in his approaching infinity. It looks like our area is getting closer and closer to pie. Right? And if you think about the area of a circle whose radius is one, which is what this polygon is sitting inside, then area is gonna equal pi our square. That&#x27;s gonna be pi times one squared. The area of the circle is pie, so this kind of makes sense.

Peter Winans · Answer

all right, We&#x27;ve been given a formula that talks about a polygon circumscribed by a circle. And we have a formula for the area. It says a equals end times the tangent 180 degrees divided by n. So we&#x27;re going to calculate the area for n equals eight to start with. Okay, in the end, of course, represents the number of sides of the polygon that&#x27;s inside this circle. So we&#x27;re starting with an eight sided polygon. So the area we&#x27;re simply going to plug the end into the appropriate place. So in this particular case, it&#x27;s gonna be eight times the tangent of 180 divided by eight. And that&#x27;s in degrees. So of course, we want to make sure before we do anything that our calculators and degree mode. So this is going to be I can just take this in right now. That&#x27;s gonna be eight times detention of 1 80 divided by eight, and we get approximately 3.13 four when we take that two. We wanted five decimal places, not three. My apologies. So we&#x27;re gonna get 3.31 31371 uh, three. Let me just double check that. 3.31 three Someone. Yep. Okay, so next we&#x27;re gonna do the same thing. Only this time we wanted to be 100. So now a is going to be 100 times two tangent, 180 divided by 100. So we&#x27;re gonna have 100 times the tangent of 1 80 provided by 100 and we get three points. 14 26 three And next you want And to be 1000. So if an is 1000 and a is going to be one 1000 times two tangent, 180 over 1000 and this is going to equal 1000 times attention of 180 divided by 1000 and we get approximately 3.14 16 01 rounded to five decimal places. And last we&#x27;re gonna try this were any is equal to 10,000. So we&#x27;re going to get 10,000 times two tangent, 180 over 10,000. So 10,000 times attention of 180. Divided by 10. Those in and we get approximately 3.14 159 Now, let&#x27;s take a look Let&#x27;s go back. All right. The number of the sides of the polygon is increasing right? When it was eight, we got an area of approximately 3.13 371 When N was up to 100 we got 3.14 to 63 When was in 1000 than three. We got 3.14160 and when and was 10,000 we got 3.14159 Now, what&#x27;s happening is we have a circle with a polygon inscribed inside it. And as the number of sides of this polygon increase, the more closely that polygons going to start to resemble a circle so as and is increasing right as an is increasing, the area seems to be approaching approximately a value of 3.14 which, if you look at when and was 10,000 we are going to be approaching a value of pie, Right? If you think about that, that makes sense. Because if we have a circle whose radius is one, an area equals pi times the radius squared, then the area is going to be pi times one squared. So we get an area of pie

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

Kevin Harmer · Answer

Okay. We&#x27;re trying to find the area of a regular regular polygon inscribed inside of a circle. No. How do we do that? Well, we know that the amount of urgencies are gonna equal the amount of triangles we have inside the circle so we can find the area of one of these triangles and then multiplied by So specifically, how do we find these triangles now? So we have the radius of our circle are and then we no, that each of the center to the Verdecia is gonna be equal to our And we know that we have advert disease, so kind of start this off. We have the area of one of our triangles and people toe 1/2 times the center to the verge ISI, which is our but then the next for to see so to ours and then sign what time is the angle between them? The angle between them has to be related to the circle. The circle goes around 360 degrees. So for an a polygon with enver disease, that 300 sixties degrees is going to be split up end times. So our angle it&#x27;s going to be 360 divided by N and I remember this is the area of a triangle inside, not the entire polygon. Whoever the area, the entire polygon is equal to this area that we have right here, which is the area of a triangle times n it&#x27;s times end because we&#x27;re gonna have an amount of triangles. Insider polygon. So the area over Pa Leon, he&#x27;s gonna be n times 1/2 so I will put that in the denominator. Many times Our times are r squared, time to sign 360 degrees divided by and and that is our answer.

Elizabeth Xu · Answer

here were given that N is equal to eight and R is equal to turn centimeters. So that means our area is you go to what half times? Eight times 10 squared terms, the sign of to pry a great And if you put them into a calculator, look at this is equal to 282.8 centimeters squared.

Problem

ANGLE OF INCLINATION Recall (Section $2-3$ ) the …

Problem 96 Hard Difficulty

Answer

$=\pi r^{2}$
$=\pi \times(1)^{2}$
$=3.14159$

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Precalculus: Graphs and Models

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$=\pi r^{2}$$=\pi \times(1)^{2}$$=3.14159$

Precalculus: Graphs and Models

Catherine R.

Anna Marie V.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

Raymond A. Barnett, Michael R. Ziegler, Karl E. ByleenPrecalculus: Graphs and Models

Catherine R.

Anna Marie V.

Maria G.

Kristen K.

$=\pi r^{2}$
$=\pi \times(1)^{2}$
$=3.14159$

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen

Precalculus: Graphs and Models