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Areas of House Lots

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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.
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Practical Geometry

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