Calculating Areas of Quadrilaterals and Triangles

Date: 08/30/2005 at 18:22:18
From: Meghan
Subject: measurement

What portion of an acre is a plot of land 85 x 174 x 73 x 118?  What
portion of an acre is a plot of land 150 x 118 x 150?

I'm not sure of the dimensions of an acre.

Date: 08/30/2005 at 20:25:30
From: Doctor Rick
Subject: Re: measurement

Hi, Meghan.

This is perhaps the most common question we get that we cannot answer!
(At least your first question.)

When you want to find the area of a quadrilateral (figure with four 
straight sides), it isn't enough to know the lengths of the four 
sides.  If you can imagine taking four sticks with lengths 85, 174, 
73, and 118 inches, and hinging them together at the ends, you'll 
find that the result is not a rigid figure.  You can change the angles 
between the sides, and the area will change as you do that.

If, in addition to the lengths of the sides, you also knew the angle 
between two of the sides, we could find the area.  (If you have a site 
plan that shows not just the lengths but the directions of the sides, 
we could find the angles between sides.)  Or if you could measure the 
length of one of the diagonals, the distance between opposite corners 
of the lot, that would be enough information too.

Now for the second question: if you're saying that this lot is a 
triangle, the lengths of the sides *are* sufficient to find the area. 
If you try the same hinged-stick experiment, you'll find that the 
triangle is rigid.

The formula to use here is Heron's formula for the area of a triangle:

  Area = sqrt(s(s-a)(s-b)(s-c))

where a, b, and c are the sides of the triangle, and

  s = (a+b+c)/2

is the "semi-perimeter" (half the perimeter) of the triangle.  With 
your numbers, we get

  a = 150
  b = 118
  c = 150
  s = (150+118+150)/2 = 209
  Area = sqrt(209*59*91*59)
       = sqrt(66205139)
       = 8136.7

If the lengths are in feet, the area of the triangle is 8136.7 square 
feet.  One acre is 43,560 square feet, so the area is

                   1 acre
  8136.7 ft^2 * ----------- = 0.18679 acre
                43,560 ft^2

If you can get a diagonal of the quadrilateral in your first question, 
you can use this same method to find the area of each of the triangles 
into which the diagonal divides the lot.  The area of the lot is the 
sum of these two areas.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/