Introduction: The Diagonal Wrapping Fallacy
Many of you have seen videos on how to diagonally wrap a present. At some point the person showing this technique will invariably say that it "saves paper". Although this technique does have some advantages, saving paper is not one of them. This Instructable will explain how to size your paper for diagonal wrapping. Included is a spreadsheet, that given the dimensions of your box to be wrapped, will calculate the required paper size for both diagonal and rectangular wrapping.
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Step 1: Two Dimensional Packages
There is really no such things as a two dimensional package. However objects like LPs, calendars, booklets, etc have such a small thickness compared to the length and width, they can be thought of as two dimensional.
A LP is 12" by 12". To cover it you will need 12" by 12" on the front and 12" by 12" on the back for total of 288 sq inches. This could be covered by conventional wrapping methods with a piece of paper 12" by 24". To diagonally wrap, measure the LP's diagonal (16.97") and cut a square with all sides this length. Either will completely cover the LP and both use 288 sq inches of paper. Of course the actual paper you use should be a bit oversized to allow some overlap for taping. I would use 18" by 18" or 13" by 25".
This step's photo shows the equations to size paper for a rectangular package. Note that if you had a flat package 12" by 6", the paper dimensions for a conventional wrap would be 12" by 12". Flip the package 90 degrees and you have the alternative dimensions of 6" by 24". In either case the dimensions for a diagonal wrap would be 13.4" by 10.7". With these three values you should be able to find a piece of paper to use.
Included with this Instructable is a spread sheet the given the length and width of the bow will give the:
Dimensions for a conventional wrapping
Dimensions for an alternative conventional wrapping.
Dimensions for a diagonal wrapping.
From these values you can choose which wrapping method works best for you specific package.
Step 2: From Two Dimensions to Three
If a box has the dimensions of y by x by z, the area to be covered is:
2yx +2yz + 2xz
Imagine if you took the box, on the right, in the photo, slit the four z length sides, and smashed it flat. You would get the two dimensional shape shown on the right. It is a rectangle (y+z) long and (x+z) wide. The area of both sides is:
2(y+z)(x+z) = 2yx + 2yz + 2xz + 2z2
Note that this method will waste 2z2 worth of paper. This is why the z dimension should be the smallest of all three.
Given this model it is time to derive the paper dimensions required to wrap a three dimensional package.
Note: As I was writing this, I made a mistake in my calculations. I must admit I was drinking while doing this. Luckily I found my error before anyone got hurt. So if you are going to drink then don't derive!
Step 3: Three Dimensional Packages
Given the shape shown in the previous steps the photo in this step shows the dimensions of the paper given the length (y), width (x) and height (z) of your package. These dimensions can be crunched with a calculator or you can go to the next step.
Step 4: Spreadsheet
Now the equations generated for step 3 are pretty complicated and I don't think I could crunch them on a calculator without making a mistake. So I wrote this spreadsheet.
I have a present waiting to be wrapped that is 9½" long, 6" wide" and 5 ¼" tall. I enter these three values and I get the dimensions for the three solutions. I decide to go with the diagonal wrap because I have a piece of paper 20" square. Note that all three of the solutions require the same amount of paper.
Now I have to wrap a basketball. It comes in a cube 10" to a side. I enter in these values and I get two solutions:
Conventional Wrapping: 20" by 40"
Diagonal Wrapping: 28.3" by 28.3"
I will also go with the diagonal wrap.
I have to wrap a pen box 7" by 2" by 1½". Plug in the numbers and I get.
Convention Wrap: 8.5" by 7"
Alternative Wrap: 3.5: by 17"
Diagonal Wrap: 9.2" by 6.5"
Normally I would pick the conventional wrap but in the case I have a sliver of paper that is the perfect for the alternative.
Lastly I have a bottle of Scotch. It isn't getting wrapped!
Step 5: Conclusion
So any of these three methods use the same amount of paper. so saying the diagonal method uses less paper is technically false. However these three options do give you a choice of paper size and you may already have a scrap that is perfect for one them. So maybe knowing these three options and being able to calculate their required paper size will allow you to use the odd pieces of scrap paper and save money.