Some shapes can be broken down into groups of simple shapes, and the areas of all pieces summed to a single number. Consider an L shape – one vertical and one horizontal rectangle – or an H; two vertical and one horizontal rectangle. Even a block A can be broken down to rectangles and triangles – three rectangles and six triangles. The more complex the shape, though, the more difficult it becomes to chop it into those simple polygons.
Fortunately, the mathematicians have developed a formula you can use to calculate the area of a polygon whether it’s irregular or not and independent of the number of sides. To use this formula, however, you must use the X-Y coordinates of all of the vertices. Depending on the size of the area in question, there are a couple of methods for getting those numbers.
| For small areas like irregularly-shaped rooms or buildings, one way to get the coordinates is to plot all the measurements to scale on graph paper. Where there are angled sides, you’ll need to dig a protractor out of the junk drawer. Once you have an outline plotted, choose a point on the graph paper to be the origin (0,0 point) of your coordinate system and use the grid of lines to assign coordinates to all the vertices. Next, build a list of the coordinate pairs in order around the polygon – clockwise or counterclockwise, it makes no difference. A simple method is to build a spreadsheet, putting the X (left-right) coordinates in one column and the Y (up-down) coordinates in a second column. Once the list is complete, copy the first pair of coordinates and add the duplicate to the end of the list: this “closes” the polygon. Now it’s time to run the calculations… |
The general formula is
Area = {(x1y2-y1x2) + (x2y3-y2x3)… + (xny1-ynX1)} / 2
In words, that means that you multiply the x coordinate of a point by the y coordinate of the next point in the list, then subtract the product of the point's y coordinate and the x coordinate of the next point. Sum all of those values for the entire list, and divide the total by two: that’s the area.
On a spreadsheet with the X coordinates in column A and the Y coordinates in column B, you can build a formula in column C to calculate the term for each pair of coordinates. Type the formula
=(A1*B2)-(B1*A2)
in cell C1, and copy that formula all the way to the next-to-last line in the file. Sum column C and divide by 2 to determine the area of the polygon.
For example, a square five units on a side has vertices at (0,0) (0,5) (5,5) and (5,0). The list of coordinates, including the repeated first pair, looks like
| 0 | 0 | |
| 0 | 5 | |
| 5 | 5 | |
| 5 | 0 | |
| 0 | 0 |
If you insert the formula above in column C, the results look like
| 0 | 0 | 0 | ||
| 0 | 5 | 25 | ||
| 5 | 5 | 25 | ||
| 5 | 0 | 0 | ||
| 0 | 0 | 0 |
And if you sum column C and divide by 2, the calculate area is 25 – the same answer you’d get multiplying 5 x 5. Of course, the simple multiplication is easier for a square, but what about the polygon below? You can see the list of vertices and the formula values out to the right; the polygon has a total area of 207.5 square units.
If the area in question is too large to accurately represent on graph paper, such as a plot of land, you can capture the coordinates with a GPS unit. Walk the boundary and set a waypoint at each vertex of the polygon. Export the waypoints to a spreadsheet in the form of UTM coordinates, and run the same procedure.
If you do use UTM coordinates, the final answer will be in square meters, which you will probably wish to convert to square feet or to acres if you’re in the USA. To convert to square feet, multiply square meters by 10.764; to convert to acres divide square meters by 4046.86
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