How can the area of a figure be calculated using its coordinates?

The method of calculating the area of a figure using its coordinates involves applying the shoelace formula (or Gauss's area formula). This technique requires taking the coordinates of the vertices of the polygon in sequential order and involves a process of cross-multiplication.

To use this method, you would list the coordinates in pairs, multiply the x-coordinate of each vertex by the y-coordinate of the next vertex, and then sum these products. The next step is to do the opposite: multiply the y-coordinate of each vertex by the x-coordinate of the next vertex and sum those products as well. The final area is determined by taking the absolute difference between the two sums, dividing by 2. This formula effectively captures the geometric relationships of the polygon's vertices, allowing for a precise calculation of the area enclosed by the figure.

The other options do not yield an accurate measure of area based on coordinate geometry principles. For instance, simply adding all the coordinates together doesn't reflect the geometrical arrangement or spatial characteristics of the figure. Averaging the coordinates does not account for the overall shape and structure of the polygon. Similarly, subtracting the highest coordinate from the lowest fails to consider the complexity of the figure's boundaries and is not applicable to area determination.