pyramid or cone
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No. Polyhedra are three-dimensional objects, and the base of any prism, strictly speaking, should be two-dimensional. Any convex polygon (which is two-dimensional) can form the base of a pyramid. A circle, which is technically not a polygon, is the only convex two-dimensional figure that cannot form the base of a pyramid because it forms the base of a cone.
Because that is how a polyhedron is defined.
A cone with included interior is 3-dimensional. However, if you are not including the interior it is a 2-dimensional surface residing in a 3-d ambient space. If you're utilizing the common topological definition of dimension, you can derive that a cone (surface only) is 2 dimensional by looking at its open sets.
2 dimensions is having only the dimensions of height and width, like a square. 3 dimensions is having height, width and depth, like a cube.
A three dimensional shape is where you have height, width, and length where as in a two dimensional shape you only have height and length