Let $f = \sum_{k=0}^n \varepsilon_k z^k$ be a random polynomial, where $\varepsilon_0,\ldots ,\varepsilon_n$ are iid standard Gaussian random variables, and let $ζ_1,\ldots,ζ_n$ denote the roots of $f$. We show that the point process determined by the magnitude of the roots $\{ 1-|ζ_1|,\ldots, 1-|ζ_n| \}$ tends to a Poisson point process at the scale $n^{-2}$ as $n\rightarrow \infty$. One consequence of this result is that it determines the magnitude of the closest root to the unit circle. In particular, we show that \[ \min_{k} ||ζ_k| - 1|n^2 \rightarrow \mathrm{Exp}(1/6),\] in distribution, where $\mathrm{Exp}(λ)$ denotes an exponential random variable of mean $λ^{-1}$. This resolves a conjecture of Shepp and Vanderbei from 1995 that was later studied by Konyagin and Schlag.