The characteristic ratio for a parabola

Area, Parabola
Facts about the area under a parabolic curve were known to the ancient Greeks; no calculus is necessary to derive these facts. Take a tangent, a, to the parabola at E, and a line, c, through E, that is parallel to the line of symmetry of the parabola. Pick a second point, G, on the parabola, and form the parallelogram with vertices E and G, and sides determined by the directions of the two lines. Then the parabola cuts the rectangle in the ratio 1:2. That is, The red region is 1/3 the area of the parallelogram. See for a way to think about this situation in terms of a coordinate system.