How maths can help you wrap your presents better

Acute solution

The method sometimes works for triangular prisms too. Measuring the height of the triangle at the end of the prism packaging, doubling it and adding it to the overall length of the box gives you the perfect length of paper to cut to cover its triangular ends with paper three times for a flawless finish.

To wrap a tube of sweets or another cylindrical gift with very little waste, measure the diameter (width) of the circular end and multiply it by Pi (3.14…) to find the amount of paper needed to encircle your gift with wrap. Then measure the length of the tube and add on the diameter of one circle to calculate the minimum length of paper needed. Doing this should mean the paper meets exactly at the centre of each circular end of the gift requiring one small piece of tape to secure it. But it’s best to allow a little extra paper to ensure the shape is completely covered or risk spoiling the surprise.

Circling back

If you have bought anyone a ball, then woe – spheres are arguably the hardest shape to wrap. It’s impossible to cover a ball smoothly using a piece of paper, not only because the properties of paper stop it from being infinitely bendable, but because of the hairy ball theorem, says Sophie Maclean, a maths communicator and PhD student at King’s College London. The theorem explains it is impossible to comb hair on a ball or sphere flat without creating at least one swirl or cowlick.

“If you think about putting wrapping paper round a ball, you’re not going to be able to get it smooth all the way round,” says Maclean. “There’s going to have to be a bump or gap at some point. Personally, I quite like being creative with wrapping and this is where I would embrace it. Tie a bow around it or twist the paper to get a Christmas cracker or a present that looks like a sweet.”

If paper efficiency is your goal when wrapping a football, you may want to experiment with a triangle of foil. An international team of scientists studied how Mozartkugel confectionery – spheres of delicious marzipan encased in praline and coated in dark chocolate – are wrapped efficiently in a small piece of foil. They observed that minimising the perimeter of the shape reduces waste, making a square superior to a rectangle of foil with the same area.