Gift Wrapping: Minimal Surface & Packing Math (3D)

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

The real math

Minimal wrapping paper. Wrap a box (L×W×H) by rolling a strip around its W×H cross-section — that strip's short side must equal the full perimeter 2W+2H (it covers, in order, the bottom, one side, the top and the other side face). Closing the two open L-direction ends needs a triangular flap folding in from each end that meets exactly at the center with no gap when its depth is (W+H)/2 — so the strip's long side is L + (W+H). Minimum paper, no overlap: (2W+2H) × (L+W+H).

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

3D bin packing. Fitting boxes into a container is the NP-hard 3D bin-packing problem — there's no known fast exact algorithm for many boxes, so real packing software uses heuristics. This simulator runs a genuine extreme-point heuristic: place the largest remaining box (trying all 6 axis-aligned orientations) at the lowest free anchor point where it doesn't overlap anything already placed, then spawn three new anchor points at its far corners. Repeat until every box is placed or none fit.

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