Suppose you have a circular cake or pizza that needs to be cut into six pieces and you don’t have a cooking protractor. How could you cut it into – at least roughly – sixths?

This is something that I’ve always done by eye, and always messed up. Until very recently, when I realised I could harness the power of geometry.

Have a think about it. Solutions below the line.


  • Cut a radius ((Finding the centre of the cake is left as an exercise))
  • Find the midpoint of the radius
  • Eyeball where the perpendicular to that radius meets the cake’s circumference
  • Cut a radius to that point
  • What you have there is a bona fide sixth of a cake.

The point on the circumference is the same distance from the centre and the end of the first radius, and the two radii are equal, so the angle must be $\piby 3$.


Thirds follow a similar idea.

  • Mark the centre of the cake, O
  • Find a point, P midway between the centre and the circumference
  • Eyeball a perpendicular to OP at P and find where it meets the circumference
  • Cut a radius from each of these points to the centre
  • Bingo, a third of a cake!

Don’t eat it all at once.