*Reading time : 10 minutes*

I bought a roll of wrapping paper.

I know, Carolineâ€¦ itâ€™s not very environmentally-friendly.Â

But now that itâ€™s done, now that I have this roll of paper, what can I do to make the best of the situation?

Itâ€™s true, Caroline, I could return it!

But before I do that, if itâ€™s OK with you, I want to try to figure out how I can wrap as many presents as possible with this roll!

So, what are the dimensions of this roll?

To begin with, letâ€™s agree that itâ€™s not the roll itself but rather whatâ€™s wrapped around it thatâ€™s of interest!

The package only provides the surface area: **\( \mathbf{12\,m^2} \)**

Thatâ€™s interesting, but it wonâ€™t help me very much given that there are infinite length/width combinations of paper with an area of **\( \mathbf{12\,m^2} \).**

Here are 2 examples:

Give me a minute. Iâ€™m going to go measure. Iâ€™ll be right back.

â€¦

â€¦

Ok, Iâ€™m back!

The width of the roll isÂ **\( \mathbf{100\,cm} \)**,Â so the length is aboutÂ **\( \mathbf{12\,m} \)**

\( \mathbf{\dfrac{12\, m^2}{100\, cm} = \dfrac{12\, m^2}{1\, m} = 12\, m} \)

In total, I have a piece of paper that is **\( \mathbf{12\,m} \)Â **long andÂ **\( \mathbf{100\,cm} \)Â **wide.

Good!Â

Now that I know that, what can I do to use the paper as efficiently as possible to wrap the presents?

First, letâ€™s say I have a **cubic present** to wrap. And to keep things simple, letâ€™s say that the measurement of one side is **1 unit (u)**.

Normally, I like to keep things as simple as possible:

I try to bring the 2 sides of paper together so that they meet at the centre of the top surface.Â

Hereâ€™s a plan view of the present placed in the centre of the wrapping paper.Â

Vertically, I want the 2 sides of the paper to meet in the middle of the top of the box, which means I need about 4 times the length of the box.Â

Horizontally, I keep about 3 times the width of the box so that the flaps cover the sides.Â Â

I know itâ€™s not ideal, but Iâ€™m trying to reproduce the classic approach of a fairly experienced gift wrapper (me!).Â

To summarize, I used a surface area ofÂ **\( \mathbf{3 \times 4 = 12\,u^2} \)Â **Â to cover the present.

Of theseÂ **\( \mathbf{12\,u^2} \)**, the surface area of the cubic box representsÂ **\( \mathbf{6\,u^2} \)**.

This meansÂ **\( \mathbf{6\,u^2} \)**Â were wasted. This is the surface area of the paper that was not used to cover the cube.Â Â

No, Caroline, thatâ€™s not great at all.

So, what can I do to improve the situation?

Wait! I think I have an idea!

Iâ€™m going to try using the net of the cube.

Even better, I just realized I can use the tool from Buzzmath to represent it!

Note that the net I chose here is just one of several possibilities!

In fact, how many nets does a cube have?

#### đź‘‡voir les dĂ©veloppements possibles đź‘‡

**11 possible nets!**

If I cut the wrapping paper to correspond exactly to the net of the cube, there will be 0 loss.Â

Z. E. R. O.Â

The ratio would beÂ **\( \mathbf{1\,:\,1} \)**. Each square centimetre of paper would be used to cover a part of the present.

Thatâ€™s amazing!

But, itâ€™s not very practical becauseâ€¦Â

- â€¦ Iâ€™ll need to use a lot of tape to attach all the sides
- â€¦ if I make a crooked cut, there will be a gap exposing part of the present
- â€¦ there will probably be lots of bits and pieces of unused paper
- â€¦ itâ€™ll take too long

Nevertheless, letâ€™s use this net:

Hereâ€™s my idea!

In comparison with the preceding image, if we consider that the central square (one side of the present) has a surface area of **1 square unitÂ (\( \mathbf{u^2} \))**, then we have a total surface area ofÂ **Â \( \mathbf{8\,u^2} \)**Â with a loss ofÂ **\( \mathbf{2\,u^2} \)**.Â

There are 2 interesting observations to be made here:

- Less paper is used by placing the cube diagonally.
- A greater proportion of the paper is used to cover the cube (present).

If you come up with an even more efficient method, please let me know!

In the meantime, Iâ€™m off to return the roll of paper!

After all, itâ€™s way more efficient to just use a pillow case!