English is constantly changing. In modern mathematics it is useful to make a very general definition for a CONE: a cone consists of all the points on all the lines that join a distinguished point, called the VERTEX, to the points of the BASE. For a (right) circular cone, the base is a circle and the vertex is a point on the line that passes through the centre of the circle and is perpendicular to the plane of the circle. But more generally, for the base we can take any curve we want, or we can take any region of the plane we want. We can even allow the base to have any dimension we want. Probably most of us picture a cone in a less general way to be the solid obtained by joining the points of a region of a plane (such as a circular disk or or an ellipse) to a vertex that is outside the plane of the base. A PYRAMID is then the special case of a cone where the base is a polygon such as a square (as in the familiar pyramids of Egypt) or a triangle (when the pyramid is called a tetrahedron). The sides of a pyramid are triangles formed by the vertex and one edge of the base polygon. The Oxford English Dictionary says that the meaning of pyramid was
"formerly sometimes extended to include the cone, which differs in having a circular (or other curved) base, and a continuous curved surface between the base and the apex",
and they provide an example from 1795. Probably few, if any, in the mathematical community today would agree with that 1795 statement.