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Considering a circle to be an infinite sided ngon 
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#19
May1114, 11:37 PM

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#20
May1214, 12:07 AM

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#21
May1214, 01:35 AM

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#22
May1214, 03:28 AM

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It is good to study a real example of approximating a circle as an infinite sided polygon.
For example, the Viete's formula: http://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula Viete's formula represents a sequence of polygons with numbers of sides equal to [itex]2^{n}[/itex], inscribed in a circle.The Viete product is: [itex]2/\pi = U_{1}/U_{2} \cdot U_{2}/U_{3} \cdot U_{3}/U{4} \cdot \cdot \cdot \cdot = U_{1}/U_{\infty}[/itex] the Viete product telescopes to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a [itex]2^{n}[/itex]gon). Alternatively, the terms in the product may be instead interpreted as ratios of perimeters of the same sequence of polygons, starting with the ratio of perimeters of a digon [itex]U_{1}[/itex],(the diameter of the circle, counted twice) and a square [itex]U_{2}[/itex] , the ratio of perimeters of a square [itex]U_{2}[/itex] and an octagon [itex]U_{3}[/itex], etc etc up to the ratio of perimeters of [itex]U_{\infty1}[/itex] and [itex]U_{\infty}[/itex] . [itex]U_{\infty}[/itex] is the perimeter of [itex]2^{\infty}[/itex]sided polygon. If the "radius" of this [itex]2^{\infty}[/itex]sided polygon is equal to 1, its diameter is equal to 2 (= [itex]U_{1}/2[/itex]), then its perimeter is equal to [itex]2\pi[/itex], hence [itex]U_{\infty}/(U_{1}/2) = 2U_{\infty}/U_{1}= U_{\infty}/2 = 2\pi/2 = \pi[/itex] this is the same result as we obtained with the Viete's formula [itex]U_{1}/U_{\infty} = 2/\pi[/itex] All the time a distinction is made between a circle and [itex]2^{\infty}[/itex]sided polygon, which is just the limiting case of [itex]2^{n}[/itex]gon. It might lead to an error to believe that a polygon transforms into a circle at an "infiniteth" step. The error just seems to disappear if we are free to call a [itex]2^{\infty}[/itex]sided polygon a circle. I used the Viete's formula from Jörg Arndt book Pi  Unleashed: http://books.google.fi/books?id=Qwwc...0holes&f=false 


#23
May1214, 04:21 AM

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#24
May1214, 04:53 PM

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I just touched on this very concept with my high school geometry class.
In order to derive the formula for the area of a circle, we assumed that a circle was an "infinitygon." Then, using the formula A=(1/2)ap (where a is the apothem and p is the perimeter) we substituted in the radius for a (since every apothem in an infinity gon is a radius) and then the circumference formula for p. We get A=(1/2)2(pi)r^2, or, pi*r^2. I warned the students that a circle is not, by definition, a polygon, but for the sake of calculating the area, it's useful to imagine that it is one since we can use what we already know to describe this new concept. I'm wondering what others think about that  isn't that how they originally calculated the area of circles? They did repeated approximations which got closer and closer to a number, which they then created a formula from? Comments would be welcome. 


#25
May1214, 05:00 PM

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In addition to that, by increasing the number of vertices of the inscribed and circumscribed polygons, and calculating the sequences of perimeters, you see that these values converge towards a single value, which will be (or what we call) the perimeter of the circle. I don't see any logical or pedagogical reason to force the students to imagine the circle as a polygon. The polygons serve as approximations, that is the whole idea. 


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