Two Hexagons from A4 Minus Square

Images are licensed under the Creative Commons Attribution-NonCommercial 4.0 International License

Have you noticed that the strip which stays behind when you cut a square from an A4 sheet of paper, is just the right size for cutting two small hexagons?

No origamist likes wasting paper and this is a great scenario for reuse. Assuming the short side of the whole sheet has the length a, the short side of the rectangle left after cutting off the square is h = a⋅(√2-1), and the wasted area is 2⋅h⋅h/2/√3 + h⋅(a-2⋅2⋅h/√3) = a⋅a⋅(√2-1)⋅(√2-1)/√3 + a⋅(√2-1)⋅(a-4⋅a⋅(√2-1)/√3) = a⋅a⋅((3-2⋅√2)/√3 + √2-1-4⋅(3-2⋅√2)/√3) = a⋅a⋅(√2-1-3⋅(3-2⋅√2)/√3) = a⋅a⋅(√2-1-(3-2⋅√2)⋅√3) ≈ 0.117⋅a² which amounts to only about 8% of the A4 sheet’s area (√2⋅a²) going to waste.

Additionally, I sometimes use the small triangular pieces left over after cutting the hexagons as bookmarks.

The whole rectangle you get left over by cutting a square from A4 paper happens to have dimensions and proportions similar to a US dollar bill, so you can use the whole rectangle for dollar-bill origami models as well.

Update: I described an improved method