Gift Wrapping: Minimal Surface & Packing Math

Box dimensions

Length L 30 cm

Width W 20 cm

Height H 15 cm

Minimum paper (no overlap)

Girth side (2W+2H)

70.0 cm

Length side (L+W+H)

65.0 cm

Paper area

4550 cm²

Box surface area

3300 cm²

Blueprint of the flat net: the four shaded bands are the box's bottom / side / top / side faces unrolled in one strip; the two end flaps close the open ends with no gap.

The real math

Minimal wrapping paper. Roll a strip around the box's W×H cross-section: the strip's short side must equal the full perimeter 2W+2H, and closing the two open ends needs a flap of depth (W+H)/2 on each end so the flaps meet exactly with no gap — giving a long side of L+(W+H). Minimum paper, no overlap: (2W+2H) × (L+W+H).

paper_w = 2W + 2H
paper_l = L + W + H

2D bin packing. Packing rectangles into a container is itself a classic 2D bin-packing / strip-packing problem. This view uses Next-Fit Decreasing Height (NFDH): sort the rectangles by height, then lay them left to right in "shelves" — starting a new shelf, stacked on top of the tallest rectangle so far, whenever the current one no longer fits the current row. It's a real, simple, well-known heuristic (not optimal, but with a proven worst-case bound), deliberately different from the 3D view's extreme-point method because a 2D shelf has no third axis to exploit.

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