Q: What is the horitontal cross section of a cube?

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Infinitely many.

Yes it can.

It will be the same shape as a square

The following are some shapes having a square cross section: a cube, a cuboid, a square pyramid.

Cube, Cuboid. Any prism with a quadrilateral cross section

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No, a cube cannot have an octagonal cross-section.

A square.

Infinitely many.

Yes it can.

Rectangle

Yes, but not a regular one.

It will be the same shape as a square

The following are some shapes having a square cross section: a cube, a cuboid, a square pyramid.

Along any edge

Cube, Cuboid. Any prism with a quadrilateral cross section

You cannot unless you know the angle of the plane which defines the cross section.

You would cut off a corner.