| If the figure is a triangle, and one side has been declared the base, then the height is the expected line. If the figure is a parallelogram, and one side has been declared the base, then the height may be taken to be a perpendicular from either of the two vertices not on the base. These are the only planar figures where heights are used in the Elements.
In the later books on solid geometry, other figures also can have bases and heights such as parallelepipeds, pyramids, prisms, cones, and cylinders. |
Different sides of a figure may be selected as the base depending on the application. In proposition XI.39 there are two triangular prisms. A triangle is chosen taken to be the base of one, while the base of the other is a parallelogram. The height of the first is a perpendicular drawn between two triangular opposite faces, but the height of the other is a perpendicular drawn between the parallelogram taken as the base and the opposite parallelogram.