arc length in polar is
here #(dr)/(d theta) = sin theta#
so # ds = sqrt{ sin^2 theta + (1-cos theta)^2 } \ d theta#
# = sqrt{ sin^2 theta + 1-2 cos theta + cos^2 theta } \ d theta#
# = sqrt(2) \ sqrt{ 1- cos theta } \ d theta#
# = sqrt(2) \ sqrt{ 1- (1 - 2 sin^2 (theta/2)) \ d theta#
# = sqrt(2) \ sqrt{ 2 sin^2 (theta/2)) \ d theta#
# = 2 \ sin (theta/2) \ d theta#
So, assuming length is of one full revolution.....
# S = 2 \ \int_0^{\color{red}{2 pi}} \ sin (theta/2) \ d theta#
# = 2 \ [ -2cos (theta/2) ]_0^{2 pi} #
# = 4 [ cos (theta/2) ]_{2 pi}^0 #
# = 4 [ 1 - (-1) ] = 8 #
