Proving turns per length on a solenoid

[anonymousphys]

Homework Statement

Prove that the number of turns per length on a solenoid is equal to 1/(diameter).

Homework Equations

A= 2(pi)(R)

The Attempt at a Solution

number of turns= length/(2pi*radius)
number of coils per length=(length/(2pi*radius))/length
number of coils per length=1/(2pi*radius)

I'm not sure why I have an extra pi. Did I do something wrong?

Thanks for any help.

 

[rl.bhat]

In the problem the diameter is the diameter of the wire, not the diameter of the solenoid.