The 12 hexiamonds can form six different convex shapes. They are shown below. The numbers indicate how many distinct tilings the shape has, allowing for rotation and reflection.
So far as I know, these shapes and tiling counts first appeared in Donald E. Knuth's The Art of Computer Programming, Volume 4B, as Problem 309b (p. 161) and its solution (pp. 499–500).
For other shapes that the hexiamonds can form, see The Poly Pages.
Last revised 2023-09-20.