# All Wrapped Up

What is the largest cuboid you can wrap in an A3 sheet of paper?

Well, we're often wrapping up presents - I find that many of them arrive in boxes that are cuboid. Remember those? A cuboid has 6 faces; each face a rectangle (including squares); each face paired with one the same on the opposite side and all the angles 90 degrees. So they might look like these:

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Now let's talk about the wrapping paper. You are probably used to using A4 paper for all kinds of things. A4 paper looks a bit like this:

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But two A4s together make A3:

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So we have:

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Which could look better as:

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Now for easiness we'll approximate the size of the A3 sheet to 40cm by 30cm.

The wrapping paper we'll be using is A3-size, like the picture above.

So, we're wrapping up a cuboid in this A3 wrapping paper.

The challenge is to find the largest cuboid that can be covered completely with the A3 sheet - you will probably want to stick to whole numbers of centimetres!

Rather than wrapping the cuboid like we might do a birthday present, let's think about it as if we're trying to have as little overlap as possible.

You might do this practically or you may use diagrams or even a spreadsheet - it is up to you.

What makes one cuboid bigger than another? You may like to talk to someone else about this!

Might it help to draw nets?

As you know the length and width of the paper, can you say anything about the length, height and width of the cuboid looking at the net you've drawn?

Perhaps using a spreadsheet would help?

Or how about making some "skeleton" cubes out of construction equipment or cubes?

This was a tricky problem, but many of you tackled it successfully. Your solutions depended on whether you assumed that you could cut the paper or not. Rachel from Charter Primary said:

I solved this problem by drawing a net.

The length of my cuboid was $38 \; \text{cm}$. The width of my cuboid was 14cm. The depth of my cuboid was $1\; \text{cm}$.

I tried to make the unused space as small as possible.

So, Rachel's solution kept the paper in one piece (as you would usually do when you wrap a present, for example).

This gives a cuboid with the maximum possible surface area ($1168 \;\text{cm}^2$).I wonder how you arrived at this solution, Rachel? How many differently-sized cuboids did you try before you discovered this was the largest? Rohaan from Longbay Primary looked at it in a different way, assuming that you could cut the paper:

We decided to find a cuboid that that had a surface area that
matched with the wrapping paper ($1200 \; \text{cm}^2$).

We made a starting estimate of a cuboid that was
$1\times20\times20$. It had a surface area of $880 \;
\text{cm}^2$.

Then we thought there could be a bigger cuboid that would fit.
We wondered by how much the cuboid's surface area would go up if we
changed its measurements from $1\times20\times20$ to
$2\times20\times20$. It went up $80 \; \text{cm}^2$.

We thought if we changed it to $3\times20\times20$ it would go
up by $80 \; \text{cm}^2$ again. It did. We went up until we
reached $5\times20\times20$ which had a surface area of exactly
$1200 \; \text{cm}^2$. It matched the surface area of the wrapping
paper.

Just to make sure it fitted, we drew up the surface area
($1200 \; \text{cm}^2$) on a piece of A3 paper. It fitted!

Did anyone try to find the cuboid with the largest volume that could be wrapped up in this paper? That's another challenge for you!

### Why do this problem?

This is a challenging problem, but suits a variety of working styles.### Possible approach

Some pupils will need to use practical equipment. Others will be able to draw nets and think about the relationship between the width, height and length of the cuboid compared with the dimensions of the paper. Others will want to use a spreadsheet to find all the different possibilities.Points that may well be discussed include:

1/ What determines the size of a cuboid? Surface area? Length? Volume?You could ask the group to investigate all of these, either by each group concentrating on a particular definition of "largest", or through all of them investigating each definition.

2/ Does it matter where on the A3 sheet the cuboid is placed? Does it have to be parallel to an edge to get the biggest cuboid? We would suggest that at first you limit the children to investigating only nets which are parallel to the paper's edges.

3/ If there is an overlap of paper, what size is it?

### Key questions

### Possible extension

Extension work may stem from asking, "I wonder what would happen if . . . ?"if the paper were A4 or A2?

if the paper was square e.g. 40 by 40, or 30 by 30?

if we were wrapping up balls, spheres?

if we were to cut the paper into two to wrap up two cuboids, which might be the same or different.

if . . . . . . .