Per the CA BPESLG’s test plan for the civil surveying exam, test takers need to know how to perform “trigonometric relationship to determine the area of a polygon” and “procedures for calculating area” when it comes to traverses. If you’re taking a course or self-studying, then you’ve probably heard of these three methods for calculating the enclosed area of a traverse/polygon:
- Divide the traverse/polygon into simple triangles and/or parallelograms.
- The double-meridian-distance (DMD) method
- The coordinate method
In my opinion, the easiest way to calculate the enclosed area of a traverse is the coordinate method, since it is simply a plug-n-chug problem (so is the DMD method, but it has more rules to follow) if you’re given coordinates or can easily resolve the polygon’s vertices into a coordinate system.
So, let’s take a look at the coordinate method formula for a 4-sided traverse with points A, B, C, and D:
What the heck? At first glance, this doesn’t seem like a simple plug-n-chug formula. Also, why is YA reintroduced at the end, and why is YD in the beginning term? The formula looks pretty daunting, but here’s a better way to visualize what’s going on in that nasty numerator (this figure was inspired from Chapter 6.8 from Cuomo, 2nd edition)
The products of the solid lines are positive, while the products of the dashed lines are negative (compare this to the previous formula to confirm). From this diagram above, you can see pretty clearly why the coordinate method is also called the “criss-cross” method, since you must go around the traverse, coordinate-by-coordinate, and multiply by the coordinates before and after by the one you are currently working on. You’ll notice that I have highlighted (XA, YA) in red and (XD, YD) in blue to illustrate this point. If you were finishing the calculation, you would have to multiply Point D’s x-coordinate with the y-coordinates from Point C, and Point A (the beginning point).
If that explanation confused you, don’t worry, there’s a a shortcut to this madness and no need to recall any formula!
(Somewhat of a) Shortcut to using the Coordinate Method
The best way to calculate an area using the coordinate method is by setting up a table for your criss-cross calculations. Instead of giving you a list of steps, I’ll show you an example:
Question: What is the area of the 4-sided traverse below?
Step 1.) Resolve the traverse or polygon into (x,y) or (N, E) coordinates: You’ll see that I was nice and gave you the coordinates already, but if you’re not given any coordinates and it’s too complicated to resolve the polygon into triangles and rectangles, make up your own coordinates for each vertex (it’s nice to label one of the points strategically as (0,0) to ease calculations).
Step 2.) Create a two-row table, with as many columns as there are unique points , plus one more (in this case 4 +1 = 5). The top row is for x-coordinates, bottom row is for y-coordinates. It really doesn’t matter which row you use, or if it’s Northing and Eastings, but what does matter is that you must write the coordinates in order, going clockwise or counterclockwise, starting at any vertex. Also, you must enter the same coordinates you started with in the last column (in this case point A was entered in the first and last columns).
Step 3.) Beginning with the first column and moving right, do a “criss-cross” calculation for the top row. In other words, multiply the first column/first row by the second column/second row, and write the answer underneath (shorthand notation for big numbers). Continue this for each column, moving right.
Step 4.) Do your “criss-cross” with the bottom row, writing the answer on top:
Step 5.) Calculate the sum of the top and the bottom. Be sure to include negative signs in your summation when you have negative coordinate(s).
Step 6.) Take the absolute value of the difference between the sum of top and sum of bottom. Divide this number by 2, and that’s your answer.
The above process worked well for me on the exam, and it’s pretty fast once you get the hang of it. Hell, as long as you set up the table, you could probably do all these calculations in one or two steps on your calculator. The table just helps you visualize the sequence.
If you liked this “shortcut”, here’s what I would recommend writing in your references to help you remember the process:
Four a four-sided (for simplicity) traverse/polygon, with points A, B, C, and D: