Problem

Continuation of Exercise $21 .$ ) Inscribe a regu…

06:13

Problem 21 Medium Difficulty

Inscribe a regular $n$ -sided polygon inside a circle of radius 1 and compute the area of the polygon for the following values of $n:$
a. 4 (square)
b. 8 (octagon)
c. 16
d. Compare the areas in parts (a), (b), and (c) with the area inside the circle.

Answer

(a) 2
(b) $2 \sqrt{2} \approx 2.828$
(c) $8 \sin \left(\frac{\pi}{8}\right) \approx 3.061$
(d) Each area is less than the area inside the circle, $\pi$. As $n$ increases, the polygon area approaches $\pi .$

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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

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'm not gonna sit here and calculate the areas you can go. Looked it up on your own. That'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'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'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're using a 16 sided polygon. Um, whatever. That's gonna look like one, Thio 3456789 10. Well, 30 40 50 60. Sorry about that, but I don't think you get the idea of what we'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'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'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'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's kind of like when we'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'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'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'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

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