Facing Up to Football 2

 Now that you know about ‘angle deficits’, you’re in a position to tackle some BIG polyhedra, starting with the ‘Truncated Icosahedron’…or common football.

All you need to know is that, at each vertex of the football, the manufacturer stitches together 1 Pentagon and 2 Hexagons…and the ‘angle deficit’ takes care of the rest!

  • Find the ‘angle deficit’ at each vertex.
  • Deduce the number of vertices needed to make the Total ‘angle deficit’ what it always is.
  • You now know the total number of Pentagon corners (one at each vertex!).
  • Five corners make one Pentagon… so how many Pentagons are there?
  • There are two Hexagon corners at each vertex…work out how many Hexagons there must be altogether.

 And there you are. 32 pieces of leather, and lots of stitching, and you have a ‘truncated’ Icosahedron (that’s the result of chopping off the corners of an ordinary Icosahedron) that bounces and rolls real well!

But an even better football is the Truncated Icosidodecahedron…so roll up your sleeves, and Tackle This Challenge:

Imagine stitching together a Square, a Hexagon and a Decagon to form each vertex of a large Polyhedron.

Follow the same procedure as last time:

Angle deficits … number of vertices … number of corners of each polygon type … number of actual polygons of each type.


How many little pieces of leather this time? Not quite double the number needed for the standard football, but nearly!


In fact, the advantage of more faces and more vertices is slightly offset by the fact that the faces aren’t all as similar as with only the Hexagons and Pentagons.

Which football bounces better? You’ll have to make one of each and try them out…


Still – amazing what you can work out, armed only with the single new fact of the ‘angle deficit’, isn’t it?

Open the File as a Word Document


Send site mail to [email protected]  or personal comments direct to with questions or comments about this web site.
Last modified: June 18, 2007