The Euler characteristic for polyhedra then requires that thye shape has no vertices! And that means it cannot be a polyhedron. I suggest you count the faces and edges again.
a square
A hexagon based pyramid.
It cannot be a simply connected 3d shape because it does not satisfy the Euler characteristic.
A Pyramid is a 7 lettered 3D shape with triangular faces.
A pentagonal cross-section prism would fit the given description
hemisphere a hemisphere is half of a sphere This is a pentagonal prism
A pentagonal prism would seem to fit the given description.
heptahedron
heptahedron
A pentagonal prism has 7 faces, 15 edges and 10 vertices
pyrimid
Hexagonal prism