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Calculating Polygon Area

Date: 01/11/2004 at 17:59:17
From: Kimberly
Subject: How do you find how many square miles are in a specific area

How can you determine the area of an unusual shape?  For example, how
would you calculate the square miles in a state?



Date: 01/11/2004 at 18:04:29
From: Doctor Jerry
Subject: Re: How do you find how many square miles are in a specific area

Hello Kimberly,

If the outline of a state is or can be closely approximated by a
polygon (a closed figure in which all sides are straight lines), then
there are at least two ways in which the area of the polygon can be
calculated.  The second one uses trigonometry, the first one does not.
Let's look at them both.

First method:

You need to know the coordinates of the vertices (corners) of the
polygon.  For a state, you could determine those with a GPS device or
through surveying.  Each vertex will have coordinates with respect to
some coordinate system, like (x1,y1), (x2,y2),...,(xn,yn), where 'n'
is the last vertex.  With that information, there is a formula for
calculating the area.  I can give you a reference to a web site that
explains the formula:

  Polygon Area
    http://mathworld.wolfram.com/PolygonArea.html 

If, for example, you have a plot of land with five vertices (x1,y1),
(x2,y2),(x3,y3),(x4,y4) and (x5,y5), the area would be

  A = (1/2)(x1*y2 - x2*y1 + x2*y3 - x3*y2 + x3y4 - x4y3 + x4*y5 -
x5*y4 + x5*y1 - x1*y5)


Second method:

This is essentially an alternate way to find the coordinates of the
vertices.  Once we have them, we follow the same formula as used in
the first method above.

Pick a vertex at which to start.  Let's call it HOME and give it the
coordinates (x1,y1) = (0,0).  Now we'll proceed from one vertex to the
next moving counterclockwise around the polygon.  

For each path from one vertex to the next, there are two measurements
we must determine; its length 'd' and its direction 'a'.  Length is
clear.  I'll discuss direction.  

We take EAST as 0 degrees and always measure counterclockwise.  So, if
the second point is north-east from HOME, the direction a1 will be
somewhere between 0 and 90 degrees.  If the second point is northwest
from HOME, the direction a1 will be between 90 and 180 degrees. 
Directions will always be between 0 and 360 degrees, including 0 but
not 360.  (Due EAST is 0 and not 360.)

So, the data for the first SIDE (from HOME to the second point) are  
(d1,a1).  The data for the second side (the distance and direction
from the second point to the third point) are (d2,a2).  The direction
a2 is found the same way we thought about the direction from HOME to
the second point.  Moving due east is 0, due north is 90, west is 180
and south is 270.  What direction takes you from point 2 to point 3?

We continue in this way.  To make the example simpler, suppose we have
five points.  We designate one of these as HOME.  There are five
sides.  For these we generate the pairs (d1,a1), (d2,a2),...,(d5,a5).

To HOME we give the coordinates (x1,y1) = (0,0).

To point #2 we give the coordinates (x2,y2) where

   x2 = d1*cos(a1)
   y2 = d1*sin(a1)

To point #3 we give the coordinates (x3,y3) where 

   x3 = d1*cos(a1) + d2*cos(a2)
   y3 = d1*sin(a1) + d2*sin(a2)

To point #4 we give the coordinates (x4,y4) where 

   x4 = d1*cos(a1) + d2*cos(a2) + d3*cos(a3)
   y4 = d1*sin(a1) + d2*sin(a2) + d3*sin(a3)

To point #5 we give the coordinates (x5,y5) where

   x5 = d1*cos(a1) + d2*cos(a2) + d3*cos(a3) + d4*cos(a4)
   y5 = d1*sin(a1) + d2*sin(a2) + d3*sin(a3) + d4*sin(a4)

Going one step further in this process should take you from point 5
back to HOME, and we can calculate the coordinates of HOME.
Theoretically, we should get (0,0) again, but there will be an
accumulation of small errors due to measurements.  If the answer is
not close to (0,0) then we probably made a calculation or measuring
mistake, so this is a good way to check our work.

Now that we have coordinates in the form (x,y) for each of the five
vertices, we can apply the formula we used in method 1.

Summary:

Obviously this process is a whole lot easier if we know the
coordinates of the vertices or can easily determine them in some way,
which lets us go directly to method 1.  But if we need to figure the
coordinates out first, method 2 gives us a way to do that which is
very straightforward, though it does require a lot of calculation. 

There are other methods as well.  For instance, sometimes you can
divide the original area into triangles or other shapes that you can
easily find the area of, then sum up all the individual areas to get
the total area.

Hope this is helpful.  Write back if you still have questions.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
College Geometry
College Triangles and Other Polygons
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons

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