Discover the maths of gift wrapping!

Have you ever got halfway through wrapping a present, and then run out of paper? Follow our foolproof method and discover the maths of gift wrapping!

You will need

A rectangular present
Wrapping paper
Sticky tape
Scissors
Pen and paper
Ruler or tape measure

Cutting the paper

First you need to cut a piece of wrapping paper the right size for your present.

Find a large, flat area to unroll the wrapping paper.
Put the present on the paper, about two centimetres from the side edge of the paper.
Gently roll the present over on to its side, away from the side edge of the paper, without sliding it. Next, roll it upside down, and then onto its far side. This is how wide the paper needs to be.
Leave a gap of about two centimetres from the far side of the present, and then make a small cut with the scissors.
Grab the front edge of the paper and bend it up to cover the near face of the present. Slide the present up or down the paper until about three-quarters of the near face is covered.
Carefully, without sliding the present, bend the far edge of the paper up and over the far face of the present. Make a mark on the paper, about three quarters of the way along the far face.
Now you’ve marked out the height and width of the paper, grab the scissors and carefully cut out the rectangle.
When you’re done, measure the paper’s side lengths and calculate its area.

Wrapping it up

Put the paper shiny side down, and put the present on top in the middle.
Bring the two sides of the paper up so they overlap on top of the present and pull them tight. Then hold them in place using sticky tape.
Make sure the present is still in the middle of the tube of paper, and then fold the top edge of paper down over the near face of the present.
Fold the two sides of the paper into the middle.
Finally, fold up the bottom edge. Hold the flap in place with sticky tape.
Repeat this procedure with the far face of the present. You’re done! Admire your wrapping.
When you’re finished, measure the dimensions of each side of the wrapped present (there are six of them). Calculate their areas and add them together. How does the surface area of the present compare to the area of the wrapping paper?

For more maths of gift wrapping, take two presents that are the same size, such as books or gift cards. Arrange them side by side, and then on top of one another. Which arrangement needs more wrapping paper?

What’s happening?

The original rectangle of paper will have more surface area than the final wrapped present. This has to be true, because the paper covers the whole present – and the paper hasn’t grown in size!

How much the surface area has shrunk depends on how much the paper overlaps. It will overlap wherever you stuck it down with sticky tape. In this activity, you cut the paper a bit bigger than needed, to make sure there were no gaps. The corners also make overlaps – you may have noticed triangular flaps forming when you folded the paper over the front and back of the present. The flaps are necessary to get the flat paper to sit neatly over the pointy corners.

Sometimes wrapping something can change its surface area. For example, a cup has an inside and an outside. Put the cup inside a tight cylindrical box, and it no longer has an inside surface; its surface area becomes smaller.

One way to save wrapping paper is to wrap multiple presents in the same parcel. It takes roughly the same amount of paper to wrap two blocks of chocolate on top of each other as it does to wrap just one block. The positioning matters – you won’t save much paper if you put the chocolate blocks side by side!

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