A "pyramid" in geometry is a solid whose base is a polygon (many sided planar figure) and a vertex (a single point where all the triangular faces meet).
The base of a regular pyramid is a polygon whose sides all have the same length.
As such a pyramid can have any number of faces.
An equilateral pyramid (with all sides - including the sides meeting at the vertex - has sides of the same length) is necessarily regular and can have 4, 5, or 6 sides if the base is a triangle, square or pentagon.
The equilateral triangular pyramid is also called a regular tetrahedron.
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four sided triangular ones have 5 sides, (if you include the bottom) 8 edges, and 5 vertices. pure triangular ones have 4 sides, (including the bottom) 6 edges, and 4 vertices.