I have just mislaid the question about the length of a spiral, but from memory the one in the diagram looked Archimedian rather than equiangular. If so, it can be given a polar equation 'r = a*theta'. The distance between successive 'turns' is 2*pi*a. The length from its origin out to a radius 'R' is:
L= (1/(2*b)) * ( R*sqrt(R*R+a*a) + a*a * log( R + sqrt(R*R+a*a) ) ) Les Cowley
