Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
A half-cube is cut into two pieces by a plane through the long diagonal and at right angles to it. Can you draw a net of these pieces? Are they identical?
What shape would fit your pens and pencils best? How can you make it?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.