Why does a concave lens have a negative focal length?

In optics, the focal length of a lens is determined by its geometry and the resulting behavior of light rays passing through it. A concave lens, which is thicker at its edges than at its center, causes parallel rays of light to diverge.12

According to the Gauss theory of lenses, focal points are defined by the intersection of refracted rays. For a concave lens, the emerging rays appear to diverge from a point located on the same side of the lens as the incoming light. Under the standard Cartesian sign convention used in the lens formula, distances measured to the left of the lens are considered negative. Because the focal point of a concave lens lies to the left of the principal point, the posterior focal length is mathematically defined as negative. This negative sign is consistent with the lens's function as a diverging element, which is also reflected in its negative diopter power.3124