Two Wheel Belt

x

2.

y

1.

z

1.

A classic trigonometry problem asks for the length of a belt wrapped around two wheels. The radii of the two wheels are

x

and

x+y

. The wheels are

z

units apart.

Let A and C be the two arcs of the belt and let B be one of the two segments joining the ends of the arcs. B is an exterior tangent to both circles. Then the lengths of A, B, and C are

2θx,

2(2x+y+z)sin(θ),

and

2(π-θ)(x+y)

.