The area of irregular closed polygons defined by coordinates may be conveniently calculated by summing the areas of adjacent trapeziums, as shown below. The area of each trapezium is calculated from the X axis. Note that with the apex coordinates defined in a clockwise direction the area between the bottom of the polygon and the x-axis is first added in (red trapeziums), then deducted (green trapeziums), leaving the correct area for the polygon, shaded blue. Other section properties, such as first and second moment of area, may be found in the same way, taking care to ensure that the formula gives a positive value for a positive😄, and negative for negative😄, for line segments above the X axis, and vice versa for segments below the X axis.
Voided shapes may be defined by connecting any external apex to one apex of the internal void, then listing the void apex coordinates in an anti-clockwise direction, then finally returning to the external apex. This procedure is shown for a regular hollow octagon below, but may also be used for any irregular shape, with any number of voids.