Areas of N-Sided Regular Polygon and Circle
Date: 02/24/2005 at 15:43:30 From: Rosanna Subject: How the general formula for polygons turns into a circle. In class we were discussing all the different possible polygon shapes of land a farmer could have with 1000m of fencing, and which one would give the largest area. We discovered that as the number of sides increased, so did the area. My teacher mentioned that if you find the general formula for area of a regular polgyon with a perimeter of 1000m, which is something like n(250000/Tan180), by manipulating it, you can turn it into the equation for the area of a circle. I wondered if you could simply explain how this is possible. I'm not sure how she got the general polygon formula, what the 250,000 is. Also she said you have to use radians instead of angles in the circle, and I am unsure how to do this.
Date: 02/24/2005 at 20:41:38 From: Doctor Rick Subject: Re: How the general formula for polygons turns into a circle. Hi, Rosanna. I can't quite duplicate what your teacher said, but here are some related ideas. If an N-gon (polygon with N sides) has perimeter P, then each of the N sides has length P/N. If we connect two adjacent vertices to the center, the angle between these two lines is 360/N degrees, or 2*pi/N radians. (Do you understand radian measure well enough to follow me this far?) The two lines I just drew, plus the side of the polygon between them, form an isosceles triangle. Adding the altitude of the isosceles triangle makes two right triangles, and we can use one of them to derive the equation sin(theta/2) = s/(2R) where theta is the apex angle (which I said is 2*pi/N radians), R is the length of the lines to the center (the radius of the circumscribed circle), and s is the length of the side (which I said is P/N). Putting those values into the equation, we have sin(pi/N) = P/(2NR) so that P = 2NR sin(pi/N) gives the perimeter of the N-gon with circumradius R. Can we see a connection between this formula and the perimeter of a circle? The perimeter of the circumcircle is 2*pi*R. As we increase N, the perimeter of the polygon should get closer and closer to this value. Comparing the two, we see 2NR*sin(pi/N) approaches 2*pi*R N*sin(pi/N) approaches pi You can check this out with a calculator, using big numbers for N such as your teacher's N=1000. If you calculate the sine of an angle in degrees rather than radians, the formula will look like N*sin(180/N) --> pi Now, let's back up and take another direction. What is the AREA of the polygon? I'll call the altitude of that triangle r (it is the radius of the inscribed circle). Then the area of the triangle is half the altitude times the base, or rs/2. The area of the polygon is N times the area of one triangle, since N triangles make up the polygon. So the formula for the area of a regular polygon is simply Area = Nrs/2 = rP/2 using the fact that P = Ns. That's a neat formula: The area of a regular polygon is half the product of the perimeter and the inradius. This relationship between perimeter and area is also true of a circle! The perimeter (circumference) of a circle is C = 2*pi*r If I take half the product of the radius and the circumference, I get rC/2 = pi*r^2 which is the area of the circle. Can I put these two lines of thought together? I've got one problem remaining: One formula uses the inradius while the other uses the circumradius. We can relate r and R by going back to that triangle: the ratio r/R = cos(theta/2) = cos(pi/N). As N increases, this approaches 1, so that in the limit r=R; there is only one radius for a circle. But because your formula involves the tangent, let's work with this. Back to the perimeter formula: P = 2NR sin(pi/N) = 2N (r/cos(pi/N)) sin(pi/N) = 2Nr tan(pi/N) so that r = P/(2N tan(pi/N)) Now Area = rP/2 = (P/(2N tan(pi/N)))P/2 = P^2/(4N tan(pi/N)) This finally begins to look like what your teacher said. If I put in P = 1000, I get Area = 250,000/(N tan(pi/1000) If you calculate the tangent of an angle in degrees rather than radians, this will be Area = 250,000/(N tan(180/1000) However, you can see that I said what I have to say about circles well before I got to this. I don't know how your teacher was going to use this to talk about circles. I hope this enlightens you a bit, anyway! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 02/25/2005 at 15:58:24 From: Rosanna Subject: Thank you (How the general formula for polygons turns into a circle.) Thank you very much, that really helped. You do a great job! - Rosanna
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