Sound like the Platonic Solids: Tetrahedron, Cube/Hexahedron, Rhomboid/Octahedron, Dodecahedron and Icosahedron, having respectively 4, 6 8 12 and 20 sides.
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A polyhedron.
An octagon is a polygon with eight sides. It might be regular or it might not.
It is called a polyhedron.
Systems of inequalities in n variables with create an n-dimensional shape in n-dimensional space which is called the feasible region. Any point inside this region will be a solution to the system of inequalities; any point outside it will not. If all the inequalities are linear then the shape will be a convex polyhedron in n-space. If any are non-linear inequalities then the solution-space will be a complicated shape. As with a system of equations, with continuous variables, there need not be any solution but there can be one or infinitely many.