Apex

An apex is the highest point (relative to the base of a figure) of certain 2D and 3D figures. More specifically, the term apex is usually used to refer to the highest vertex opposite the base of a geometric figure. Below are a few examples of geometric figures and their apices:

The reason that the term apex is only used to describe certain geometric figures is because many geometric figures have multiple points that would be considered an "apex." Thus, the term apex is only really used in geometric figures such as cones, pyramids, triangles, and other figures in which there is only one highest vertex opposite the base of the figure. Otherwise, it would not be clear what point is being referred to as the apex.

For example, in a rectangular prism, based on the triangle definition of an apex, we could technically refer to 4 separate points relative to a given base as an apex. The red points in the figure below represent 4 "apices" of the cube.

Furthermore, each of the 4 points in the prism has the same height as any point on the face of the prism opposite the chosen base, so none of the apices would fit the definition of being the "highest" point, since they would all be the highest point of the prism, and technically any point opposite the referenced base would be an "apex."

Apex and height

The apex of a figure is often associated with the figure's height. The height of a figure can be generally defined as the perpendicular distance from the base of the figure to the side opposite the base. If the figure has an apex, the height of the figure is the perpendicular distance from the base of the figure to the apex, as shown in the figure below.

Thus, when a figure has an apex, the height of the figure includes the apex of the figure, but the apex just describes the highest point, while the height is a measure of distance.