In the plane, the Crofton formula states that for a rectifiable plane curve $\gamma$, we have
$\int |line \cap \gamma| d\Omega_1=2\times length(\gamma)$
where $d\Omega_1$ is the translation/rotation-invariant measure on the space of all (undirected) lines.
The proof of this is to use addivity, break into line segements, and conclude that the integral is always a constant multiple of the curve length, and therefore we can just do the computation using $\gamma$ as the unit circle to figure out the constant.
Now, this logic carries over in higher dimension. Namely, suppose we are in $\mathbb{R}^3$, then
$\int |plane \cap \gamma| d\Omega_2 \propto length(\gamma)$
However, how can the constant of proportionality be determined? What would be the "easiest" choice of $\gamma$?
Remark: I have seen $\mathbb{R}^3$ versions where $S$ is a surface, and the same logic gives $\int |line \cap S| d\Omega_1 \propto surfacearea(S)$ and the constant of proportionality is easily determined using $S$ being the hollow unit sphere. But changing to curve and plane is suddenly much more difficult...