  In protoplanetary discs and planetary atmospheres, dust grains coagulate to
form fractal dust aggregates. The geometric cross-section of these aggregates
is a crucial parameter characterizing aerodynamical friction, collision rates,
and opacities. However, numerical measurements of the cross-section are often
time-consuming as aggregates exhibit complex shapes. In this study, we derive a
novel analytic expression for geometric cross-sections of fractal aggregates.
If an aggregate consists of $N$ monomers of radius $R_0$, its geometric
cross-section $G$ is expressed as \begin{equation} \frac{G}{N\pi
R_0^2}=\frac{A}{1+(N-1)\tilde{\sigma}}, \nonumber
  \end{equation} where $\tilde{\sigma}$ is an overlapping efficiency, and $A$
is a numerical factor connecting the analytic expression to the small
non-fractal cluster limit. The overlapping efficiency depends on the fractal
dimension, fractal prefactor, and $N$ of the aggregate, and its analytic
expression is derived as well. The analytic expressions successfully reproduce
numerically measured cross-sections of aggregates. We also find that our
expressions are compatible with the mean-field light scattering theory of
aggregates in the geometrical optics limit. The analytic expressions greatly
simplify an otherwise tedious calculation and will be useful in model
calculations of fractal grain growth in protoplanetary discs and planetary
atmospheres.
