Dialing a number on my phone

Geometry Level 2

If the buttons of a telephone are one centimeter away from each other, measuring from center to center.

When dialing 5927018 5927018 , what is the distance in cm that the finger travels?

5 ( 3 + 2 ) + 2 2 \sqrt { 5 } \left( 3+\sqrt { 2 } \right) +2\sqrt { 2 } 2 5 ( 1 + 2 ) + 2 2 2\sqrt { 5 } \left( 1+\sqrt { 2 } \right) +2\sqrt { 2 } 5 + 2 ( 2 + 3 5 ) \sqrt { 5 } +\sqrt { 2 } \left( 2+3\sqrt { 5 } \right) 2 5 + 2 ( 3 + 5 ) 2\sqrt { 5 } +\sqrt { 2 } \left( 3+\sqrt { 5 } \right)

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