An octagonal pyramid is an obvious answer but there are infinitely many variations where the base is surrounded by numerous layers of polygons in bands.
For example, an octagonal base, with eight rectangular sides at right angles to the base, topped by eight triangles would be like a stretched pyramid. The rectangular sides can be replaced by trapeziums and no longer be required to be at right angles to the base.
A pyramid has 1 square base.
square base pyramid
Spherical
legnthXwidthXheight Top is close, but only for a rectangular prism :(... It is actually (base area) X height. For pyramids, it is 1/3(base area) X height.
A cone perhaps?
Octagonal prism
Octagonal pyramid
cube square prism
A pyramid has 1 square base.
the cone.
pentagonal pyramid
cone
square base pyramid
Spherical
A pyramid with an n-sided base will have n + 1 vertices, n + 1 faces, and 2n edges.
A hemisphere
Sphere