  We present the first calculation of the electromagnetic form factor of the
$\pi$ meson at physical light quark masses. We use configurations generated by
the MILC collaboration including the effect of $u$, $d$, $s$ and $c$ sea quarks
with the Highly Improved Staggered Quark formalism. We work at three values of
the lattice spacing on large volumes and with $u$/$d$ quark masses going down
to the physical value. We study scalar and vector form factors for a range in
space-like $q^2$ from 0.0 to -0.1 $\mathrm{GeV}^2$ and from their shape we
extract mean square radii. Our vector form factor agrees well with experiment
and we find $\langle r^2 \rangle_V = 0.403(18)(6) \,\mathrm{fm}^2$. For the
scalar form factor we include quark-line disconnected contributions which have
a significant impact on the radius. We give the first results for SU(3)
flavour-singlet and octet scalar mean square radii, obtaining: $\langle r^2
\rangle_S^{\mathrm{singlet}} = 0.506(38)(53) \mathrm{fm}^2$ and $\langle r^2
\rangle_S^{\mathrm{octet}} = 0.431(38)(46) \mathrm{fm}^2$. We discuss the
comparison with expectations from chiral perturbation theory.
