Finding Areas of Different PolygonsDate: 09/02/97 at 16:49:08 From: alex Subject: Geometry area working out on polygons Could you please tell me how to work out the area for an equilateral heptagon, octagon, nonagon, decagon, unedecagon, and dodecagon? Thank you. Date: 09/08/97 at 15:59:36 From: Doctor Rob Subject: Re: Geometry area working out on polygons For any regular n-gon, draw it and find the center of its circumscribed circle, which we will call O. Draw radii from O to two adjacent vertices of the n-gon, A and B. Then angle AOB will have measure 2*Pi/n. Drop a perpendicular from O to AB, bisecting it at C. Then the area of the entire n-gon will be n times the area of triangle OAB, or 2*n times the area of triangle OAC. The base of triangle OAC is AC, whose length is s/2 (s is the side length of the n-gon). The altitude of triangle OAC is OC, whose length is (s/2)/tan(Pi/n), since angle AOC is half of angle AOB, or half of 2*Pi/n, or Pi/n. Thus the area of triangle OAC is (1/2)*(s/2)*(s/2)/tan(Pi/n), and the area of the n-gon is A = (n*s^2/4)*cot(Pi/n). For n = 8, 10, and 12, the cotangent can be found in terms of radicals. For n = 7, 9, and 11, it cannot. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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