Three wheels each of radius 1 have their centers at respective verticies of an equilateral triangle of side length 4. A belt is wrapped continuously around the wheels. Find the length of the belt.

L  = 6 * ( π + 1 )

Explanation:

The side of the equilateral triangle is  4, and each one of the circles is of radius 1. Then

Triangle vertex   A  B  and C

Trajectory of the belt, beginning in vertex A

1.-First circle  A  one turn

L₁  = 2*π*1   = 2*π

2.-Length between the circle A and B

L₂  = 2

3.- To wrap this circle we need to wrap the circle and to run the belt through the radius twice. The firs over the side AB and the second over the side BC, therefore

L₃ = 2*π*1 + 2*(1)

L₃ = 2*π + 2

4.-Length between two nearest points of circles B and C is 2 and length of the circle in C is 2*π*1. Then

L₄  = 2 + 2*π

Total length of the belt  is:

L  =  L₁ + L₂ + L₃ + L₄

L = 2*π + 2 + (2*π + 2 ) + ( 2 + 2*π )

L  = 6*π + 6

L  = 6 * ( π + 1 )

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