1 | initial version |

You can construct the 3-dimensional cube as follows:

```
sage: P = polytopes.n_cube(3) ; P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices
sage: P.vertices()
(A vertex at (-1, -1, -1),
A vertex at (-1, -1, 1),
A vertex at (-1, 1, -1),
A vertex at (-1, 1, 1),
A vertex at (1, -1, -1),
A vertex at (1, -1, 1),
A vertex at (1, 1, -1),
A vertex at (1, 1, 1))
```

Then construct the polyhedron with the last vertex removed:

```
sage: Q = Polyhedron(P.vertices()[:-1]) ; Q
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 7 vertices
sage: Q.vertices()
(A vertex at (-1, -1, -1),
A vertex at (-1, -1, 1),
A vertex at (-1, 1, -1),
A vertex at (-1, 1, 1),
A vertex at (1, -1, -1),
A vertex at (1, -1, 1),
A vertex at (1, 1, -1))
Then compute its volume:
sage: Q.volume()
20/3
```

In our case, the length of the edge is `2`

. If you want to replace it by `a`

, it suffice to multipliy the result by by (a/2)^3:

```
sage: a = var('a')
sage: Q.volume()* (a/2)^3
5/6*a^3
```

2 | No.2 Revision |

You can construct the 3-dimensional cube as follows:

```
sage: P = polytopes.n_cube(3) ; P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices
sage: P.vertices()
(A vertex at (-1, -1, -1),
A vertex at (-1, -1, 1),
A vertex at (-1, 1, -1),
A vertex at (-1, 1, 1),
A vertex at (1, -1, -1),
A vertex at (1, -1, 1),
A vertex at (1, 1, -1),
A vertex at (1, 1, 1))
```

Then construct the polyhedron with the last vertex removed:

```
sage: Q = Polyhedron(P.vertices()[:-1]) ; Q
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 7 vertices
sage: Q.vertices()
(A vertex at (-1, -1, -1),
A vertex at (-1, -1, 1),
A vertex at (-1, 1, -1),
A vertex at (-1, 1, 1),
A vertex at (1, -1, -1),
A vertex at (1, -1, 1),
A vertex at (1, 1, -1))
Then compute its volume:
sage: Q.volume()
20/3
```

In our case, the length of the edge is `2`

. If you want to replace it by `a`

, it suffice to multipliy the result by by (a/2)^3:

```
sage: a = var('a')
sage: Q.volume()* (a/2)^3
5/6*a^3
```

If you want to have a plot of the polyhedron `Q`

, just do:

```
sage: Q.plot()
```

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