NAME: Jacqui
GRADE: 11th
WHO: Student

Okay, I know this will sound like homework help but this question was asked in class today ad although I had the teacher re-explain it to me I still struggle to see how the answer was produced. Can you please set it out in the easiest form possible about how I could reach a suitable answer. Geometry has never been my strong point.

A soccer ball is made up of hexagons and pentagons with the same side lengths.
A manufacturer wants to produce a ball of certain diameter. The questions that follows is what side length for the polygons will produce a ball of certain diameter.

You are required to develop the general formula for the diameter (d) of the ball in term of the side length (l) of the polygons.
List any assumptions made

The assumptions being made is that there are 12 pentagons and 20 hexagons
Okay, I know the surface area of a sphere or radius is A= 4pieR²
i need to find a simplistic way to express the area of a regular hexagon of side length.

Is there any ways you could express a simplistic way for me to find out how to write a formula for this question.

Jacqi,

The technical name for a soccer ball is a truncated icosahedron. If I really had to compute the parameters, I would start with an icosahedron and chop off the vertices. Much easier, though, is to look it up. My source says the radius equals sqrt[(29 + 9*sqrt[5]) /2] when the side of the truncated icosahedron (with flat faces) is 2. The expression for the length of the circular arcs on the surface of the ball will be slightly larger (and a rather ugly expression to write out).

Chris