Areas of House Lots
Date: 02/18/98 at 14:41:45 From: Jack Redfearn Subject: Area of 4-sided figure We design and construct sanitary sewers for people in Kansas City. After the job is complete we have the task of determining sanitary sewer assessments of properties based on the square feet of their lots. Many lots are 4-sided but do not have any parallel lines. We would like to find a powerful formula that can calculate the area of a lot given the length of each side. I believe a figure is "defined" when the 4 sides are given, so there must be a way to calculate the area. However, we have been unable to come up with anything. Again, no sides are parallel and we do not know any angles.
Date: 02/18/98 at 16:15:25 From: Doctor Rob Subject: Re: Area of 4-sided figure Sorry, but the figure is not "defined" when the four sides are given. You need one more datum. It may be an angle, it may be the length of a diagonal, or some other quantity. Even if all the sides are equal, you can have a square, or a rhombus (a parallelogram), and the area of the rhombus is always less than that of the square. The acute angle in the rhombus can be anything between 0 and 90 degrees. As a result, there is no such formula. If you know the sides are a, b, c, and d, running around the boundary, and the diagonal of length e cuts the lot into two triangles of sides a, b, e, and c, d, e, respectively, then the formula to compute the area is as follows. Let s = (a+b+e)/2 and t = (c+d+e)/2. Then the area is A = Sqrt[s*(s-a)*(s-b)*(s-e)] + Sqrt[t*(t-c)*(t-d)*(t-e)]. If you know the angle X between sides whose lengths are a and b, then the Law of Cosines tells that e^2 = a^2 + b^2 - 2*a*b*cos(X). Then you can figure out e and use the previous formula. -Doctor Rob, The Math Forum Check out our web site http://mathforum.org/dr.math/
Date: 02/19/98 at 12:41:03 From: Anonymous Subject: Re: Area of 4-sided figure Thanks for your prompt reply. A couple of us in the office are working on this and we appreciate your help and suggestions. I think there will be times when we will know an angle (or be able to find out an interior angle from the subdivision plat) and/or be able to compute a diagonal. Anyway, thanks again.
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