= Solution
A three-dimensional <self-avoiding walk> has swollen size $R\sim bN^\nu$, with $\nu\simeq0.59$; the three-dimensional <Flory approximation> gives the nearby estimate $3/5$. Isotropic projection gives $\langle R_{\rm proj}^2\rangle=(2/3)\langle R^2\rangle$, preserving this exponent. For the mean distance rather than its root mean square, the orientation average instead gives $\langle R_{\rm proj}\rangle=(\pi/4)\langle R\rangle$. Neither prefactor changes its position between the ideal $N^{1/2}$ walks and the planar swollen $N^{3/4}$ walk.
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