You want [`Hexahedron`](https://reference.wolfram.com/language/ref/Hexahedron.html), which specifies a geometric region with six quadrilateral sides via a list of vertices.  Note that the vertices need to be ordered in a specific way, described in the documentation;  in your case, though, it's just the "obvious" ordering.

``` 
reg = Hexahedron[{a, b, c, d, a1, b1, c1, d1}]
Volume[reg]
Graphics3D[reg]

(* 7/Sqrt[6] *)
```
[![enter image description here][1]][1]

You can further use built-in Mathematica functions to calculate things like centroids, nearest points, etc.  Examples of this can be found in the above-linked documentation page.

  [1]: https://i.sstatic.net/Uig7v.png