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/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.