This page is intended to explain the notation used in the analytic derivation of the area formula for midpoint polygons.

We identify vectors and points, for example (x, y) = X, as illustrated below:

Now the formula for the area of a triangle can be expressed in vector operations on the vectors associated with the triangle's vertices. Consider a triangle with one vertex at the origin and its other vertices at (x₁, y₁) = X₁ and (x₂, y₂) = X₂ (see figure 7). The oriented area of this triangle is given by the dot product of (1/2)(X₁ x X₂) with (0, 0, 1), as explained in the first page on oriented area. We do not usually write in the dot product, but leave the area expressed as a vector, since this does not change the calculations.

It should be emphasized that the use of vectors here is just a notational convenience, and is not required for the derivation of the formula: it would be possible, if somewhat laborious, to arrive at the same conclusion from calculations involving solely the coordinates of the points.