Area by coordinates method

When the coordinates (northing $N$ and easting $E$, or $x$ and $y$) of the stations of a closed traverse or boundary are known, the area is found directly from them.

Formula

Number the stations $1, 2, \dots, n$ in order round the figure (clockwise or anticlockwise), with the coordinates $(x_i, y_i)$ and $(x_{n+1}, y_{n+1}) = (x_1, y_1)$.

$$\text{Area} = \frac{1}{2}\left|\sum_{i=1}^{n} \left(x_i\,y_{i+1} - x_{i+1}\,y_i\right)\right|$$

or, in the form with the coordinate differences,

$$\text{Area} = \frac{1}{2}\left|\sum x_i\,(y_{i+1} - y_{i-1})\right|$$

Procedure

1. Compute the adjusted latitudes and departures, and the coordinates of every station from a convenient origin (preferably so that all coordinates are positive).
2. Write the coordinates in order in two columns and repeat the first station at the end.
3. Multiply each $x$ by the next $y$ (downward products) and each $y$ by the next $x$ (upward products).
4. Take the difference of the two sums and halve it. The sign depends on the direction of travel; the numerical value is the area.

Example

Stations: A (0, 0), B (80, 30), C (100, 100), D (20, 90) in (E, N).

$$\text{Area} = \frac{1}{2}(17000 - 5000) = 6000\ \text{m}^2$$

The method is exact for a polygon and is suitable for a computer or calculator, with no need for dividing the figure into parts.