Summary: Anomalously strong finite-size effects have been observed for the mean square radius of gyration \(R^2_K\) of Gaussian random polygons with a fixed knot \(K\) as a function of the number \(N\) of polygonal nodes. Through computer simulations with \(N<2000\), we find that the gyration radius \(R^2_K\) can be approximated by a power law, \(R^2_K \sim N^{2\nu_K^{\text{eff}}}\), for several knots, where the effective exponents \(\nu_K^{\text{eff}}\) are larger than 0.5 and less than 0.6. Furthermore, a crossover occurs for the gyration radius of the trivial knot, when \(N\) is roughly equal to the characteristic length \(N_c\) of random knotting. Assuming an asymptotic fitting formula, we also discuss possible asymptotic behaviours for \(R_K^2\) of Gaussian random polygons.