= Solution
Let $A$ be the initial plan <area> and take compression normal to the sheet. Smooth plates exert no lateral traction, so the uniaxial <stress> is $\sigma_z=-F/A$ and the two lateral <strains> are $\varepsilon_x=\varepsilon_y=\nu F/(EA)$. To first order in <strain>,
$$
\Delta A=A(\varepsilon_x+\varepsilon_y)=\frac{2\nu F}{E}=\frac{2(0.48)(200)}{50\times10^6}=\boxed{3.84\times10^{-6}\,\mathrm{m^2}=3.84\,\mathrm{mm^2}.}
$$
This <area expansion under uniaxial compression> is the increase of the sheet's plan <area>, equivalently the <area> of either broad face. Counting the sum of both broad faces would give twice this change; edge <area> would require thickness information. The cancellation of $A$ is a small-strain linear-elastic result, not a statement that arbitrarily large compressions need no geometry data.
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