Enter the box size and see the sheet to cut, the diagonal method that can save paper, the ribbon and how many gifts one roll covers.
The calculator gives the sheet size for boxes, cubes, cylinders and round items, and counts the rolls for a whole list of gifts. In the standard method the sheet width is the wrap-around plus the seam, 2 x (width + height) + 3 in, and the length is the gift length plus twice the height plus 4 in to fold the ends; a 14 x 8 x 5 in shoe box needs 29 x 28 in (How to Wrap a Present). The calculator tests all three axes and keeps the one that uses the least paper. In the diagonal method the box sits at 45 degrees on a square sheet with side equal to the square root of 2 x (longest edge + shortest edge), a derivation published by the NRICH project at the University of Cambridge, plus a 1 in allowance; the tests check by sampling points that this sheet covers every point of the box. Cylinders use circumference + 2 in by (height + diameter + 3 in), and round items a square of circumference + 4 in. A cross-tied ribbon is 2 x length + 2 x width + 4 x height plus a 12 to 18 inch bow. For many gifts, the sheets are packed in shelves across the roll width, turned 90 degrees when needed, and the result says how many rolls to buy. The diagonal method does not always save paper: with the default allowances it beats the standard method by 41% on a cube and by 10% or less on flat boxes, and a tightly trimmed seam can reverse that on thin boxes.
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For a 14 x 8 x 5 in shoe box, a 29 x 28 in sheet with the standard method. The width is 2 x width + 2 x height + 3 in (16 + 10 + 3) and the length is length + 2 x height + 4 in (14 + 10 + 4). With the diagonal method a square 28 3/8 in on a side is enough, and it also covers the whole box.
The source formula uses 3 in (7.6 cm) and the quick method says 2 to 3 in.
28 3/8 in x 28 3/8 in
Diagonal method, 5.56 sq ft of paper.
Rule: side = square root of 2 x (longest edge + shortest edge + 1 in allowance), with the box turned 45 degrees (How to Wrap a Present, NRICH, University of Cambridge).
29 in x 28 in
Needs a roll at least 28 in wide.
28 3/8 in x 28 3/8 in
Needs a roll at least 28 3/8 in wide.
Uses 1% less paper than the standard method.
78 7/8 in
From 75 7/8 in to 81 3/4 in, depending on the bow size.
Rule: 2 x length + 2 x width + 4 x height + a 12 to 18 in bow.
The gift sits at 45 degrees in the center of the square sheet and the four corners fold to the middle of the lid.