A cone has a circular base, and sides that converge on a single point at some distance (defined as the height of the cone) above that circle. If that point is directly above the center of the circle, the cone is a right cone. In common usage, a cone is generally understood to be a right cone unless otherwise specified. The volume of a cone = 1/3 (pi)r^2(h) where r = the radius of the circular base and h = height of the cone. The surface area = pi * r * √(r^2 + h^2) + the surface area of the circular base, which is pi * r^2.
A prism is a polyhedron with two congruent parallel bases, each of which are polygons, separated by a distance, h, and sides that are parallelograms. Each vertex on one of the bases is connected by a straight line to the corresponding vertex on the other base. Prisms are named for the type of polygon forming the bases. The simplest prism is a triangular prism, with triangles for its two bases, but there is no limit to the number of sides on the bases. Simple methods exist to calculate the area for a polygon with any given number of sides. The volume of a prism is equal to the area of one of the bases (both are identical and have the same area) multiplied by h. The surface area is equal to the perimeter of a base multiplied by h, plus the area of the two bases themselves.
A cross section at any point on a prism, cutting parallel to the two bases, would be identical in size and shape to the two bases. Cutting a cone in the same manner would yield the same shape as the base -- a circle -- but the size would diminish as the distance from the base increased. If you were to completely slice off the top portion of a cone, you'd be left with a new type of three-dimensional figure, a conical frustum. Doing the same thing to a prism would leave the same type of prism, but with a smaller height.
Cutting cross sections of a cone at different angles will produce the conic sections: circle, ellipse, parabola and hyperbola (which assumes that you are cutting a double cone). The ancient Greeks studied these more than 2,000 years ago, but it was not until Rene Descartes invented analytic geometry that mathematicians were able to examine these shapes in numerical terms, without reference to sections of a cone. Conic sections are extremely important to modern mathematics and applied science. Sectioning prisms is possible, but has far fewer applications.