  The small-angle scattering (SAS) from the Cantor surface fractal on the plane
and Koch snowflake is considered. We develop the construction algorithm for the
Koch snowflake, which makes possible the recurrence relation for the scattering
amplitude. The surface fractals can be decomposed into a sum of surface mass
fractals for arbitrary fractal iteration, which enables various approximations
for the scattering intensity. It is shown that for the Cantor fractal, one can
neglect with a good accuracy the correlations between the mass fractal
amplitudes, while for the Koch snowflake, these correlations are important. It
is shown that nevertheless, the correlations can be build in the mass fractal
amplitudes, which explains the decay of the scattering intensity $I(q)\sim
q^{D_{\mathrm{s}}-4}$ with $1 < D_{\mathrm{s}} < 2$ being the fractal dimension
of the perimeter. The curve $I(q)q^{4-D_{\mathrm{s}}}$ is found to be
log-periodic in the fractal region with the period equal to the scaling factor
of the fractal. The log-periodicity arises from the self-similarity of sizes of
basic structural units rather than from correlations between their distances. A
recurrence relation is obtained for the radius of gyration of Koch snowflake,
which is solved in the limit of infinite iterations. The present analysis
allows us to obtain additional information from SAS data, such as the edges of
the fractal regions, the fractal iteration number and the scaling factor.
