i think the answer is a cone because it has a flat bottom
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Yes Vertex(Vertices) are corners or points of 3D shapes.
Only a square and a rhombus will have all its diagonals bisecting vertices. In other shapes some - but not all - diagonals can bisect vertices.
There two different shapes that have more than one vertex. One is a triangle it is a plane figure that has three straight sides and three angles and a quadrilateral which a four sided figure.
There are infinitely many shapes. Among them are polyhedra, with four or more vertices.