Formula for Area of Any Regular PolygonDate: 03/01/98 at 20:42:54 From: Andrew Miklas Subject: 3D Geometry Is there a formula to calculate the area of a regular polygon that uses the number of sides and the length of a side as the inputs? For example, Sides Length of Side Area ____________________________________ 3 6 18 4 8 64 5 2 ? 10 13 ? 100 25 ? If so, it would follow that there is a formula that calculates the volume and surface area of such a figure (when it is put into prism form), where x = formula l = prism length s = side length n = number of sides x * l = volume s * l * n + 2x = surface area Am I correct? Date: 03/02/98 at 17:04:27 From: Doctor Sam Subject: Re: 3D Geometry Andrew, You are correct that such a formula exists, and that the total surface area of a regular prism is given by your formula. But the numbers in your table are not all correct. The area of an equilateral triangle of side 6, for example, is 9 * sqrt(3), not 18. The formula that you are looking for is derived by taking your regular polygon and dividing it up into triangles in a special way. A regular polygon can be inscribed in a circle . . . just imagine the center of the polygon as the critical point P. Connect P to each of the n vertices of the polygon . . . this divides it into n isosceles triangles. The height of each triangle needs to be computed using trigonometry. Since there are n triangles, the vertex angle of each triangle (at P) is 360/n degrees. The altitude of the triangle bisects the angle, giving an acute angle of 180/n degrees. This lets us compute the altitude h as the tangent of 180/n. The side opposite the angle is half the base (half the side of the polygon) and the adjacent side is the unknown height. This gives: tan(180/n) = s/(2h) or h = s/(2 tan(180/n)) The area of this isosceles triangle is, therefore, A = (1/2) * s * h = s^2/(4 tan(180/n)) Since there are n triangles, the area of the n-sided regular polygon is x(s) = n s^2/(4 tan(180/n)) Notice that if n=4, this reduces to the area of a square, since tan(45) = 1; and if n=3, this reduces to the area of an equilateral triangle, since tan(60) = sqrt(3). I hope that helps. -Doctor Sam, The Math Forum Check out our web site http://mathforum.org/dr.math/ |
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