Thomas and Ken-Ree thought the smallest rectangle can be found by drawing around the net of the box. Here are Ken-Ree's diagrams.
Camille tried something similar but moved parts of the net around to get a smaller rectangle.
I chose a box with 7.8cm length, 5.6 cm width and 3 cm height. Firstly, I started by turning the box on a sheet of white paper drawing around it every time to get a net. Next I cut it out. Then on another piece of paper, using what I cut out, I drew the rectangle just around its borders. On this rectangle I turned the piece of paper to see what I could move to another place to make the rectangle smaller. The final rectangle measured 14.3 cm by 14.5 cm.
Well done to John from Nagoya International School who used algebra to work out this problem.