= Solution
The height is constant along each circular ring, whose <area element> after angular integration is $2\pi r\,dr$. The <volume integral> in <cylindrical coordinates> is therefore
$$
V=2\pi\int_0^R rR e^{-r/R}\,dr=2\pi R^3\int_0^1ue^{-u}\,du.
$$
Here $u=r/R$. Using <integration by parts>, $\int_0^1ue^{-u}\,du=[-(u+1)e^{-u}]_0^1=1-2/e$. Hence
$$
\boxed{R=\left[\frac{V}{2\pi(1-2/e)}\right]^{1/3}.}
$$
The dimensionless coefficient is positive, and the cube root of a <volume> has the required <physical dimension> of <length>.
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