If it a right cone then it is a circle, otherwise an ellipse.
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Circle
By definition, the circular cross-section of a cone changes linearly in width as you go along its axis. By definition, the cross-section of a prism is constant along its axis. So, by definition, a cone prism is an impossible shape.
For a right cone, it is a hyperbola which becomes and isosceles triangle when the section passes through the apex.
Pyramid, triangular prism, cone.
Then the cross-section is a circle or a point.