Look at this icosahedron

Geometry Level 3

A normal soccer ball is in the shape of a truncated icosahedron. It has 12 pentagonal faces and 20 hexagonal faces. How many edges does it have?

80 160 90 190

Estella Z. by Estella Z. , 17, USA

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1 solution

Kb E
Oct 29, 2017

According to Euler's polyhedron formula, V − E + F = 2 V-E+F=2 . Notice that the truncated icosahedron does not have any pentagons that share a face nor does it have any vertices that are only surrounded by hexagons so the number of vertices is V = 12 ⋅ 5 = 60 V = 12\cdot 5 = 60 . The number of faces is simply F = 12 + 20 = 32 F = 12+20=32 . Then by Euler's polyhedron formula, 60 − E + 32 = 2 ⟹ E = 90 60-E+32=2 \implies E = 90 .

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