A soccer ball has face which are pentagons and hexagons whic

A soccer ball has face which are pentagons and hexagons, which each pentagon surround by hexagons. Assuming that every vertex has degree 3, dertermine the nuber of vertices, edges, and faces on the soccer ball.

Solution

An icosahedron is a platonic solid having 20 congruent equilateral triangular faces (F=20), 30 edges (E=30) each five meets at each vertex & 12 identical vertices (V=12

) lying on the spherical surface (with certain radius).

When an icosahedron is (partially) truncated at all its 12 vertices then each of 20 triangular faces become a hexagon & each of 12 vertices produces a new petagonal face. Thus a truncated icosahedron has 12 regular pentagons & 20 regular hexagons.

While the process of truncation produces 12×5=60

new vertices & 12×5=60 new edges in addition to its 30 original edges. Thus it has total 60+30=90 edges. For a truncated icosahedron, F=12+20=32, E=90 & V=60 which duly satisfies Euler\'s formula (F+V=E+2

). It is also called Archimedean solid.

A soccer ball is analogous to a truncated icosahedron. Hence it has 12 (regular) petagons & 20 (regular) hexagons.