| # Surface Area Consider a surface in that is described by a function from the square to . The corresponding surface area is We can obtain this formula using a grid of the square , with grid points , , where is a small increment. With this discretization, to a point on the grid, we associate a rectangle on the plane . The area of the rectangle is , with the spacing of the grid. However, the area of the corresponding surface is not the same, as it is not a rectangle but a parallelogram. This parallelogram is defined by its two adjacent vectors The formula for the area of a parallelogram defined by two adjacent vectors is the magnitude of the cross-product between the two vectors: where is the angle between the two vectors. Using the expressions for given above, after some algebra we obtain the formula for the area of the parallelogram: which proves the formula above.