  A clean determination of the angle $\alpha$ of the unitary triangle from
$B\to \pi\pi$ decays requires an isospin analysis. If the $B \to \pi^0\pi^0$
and $\bar B \to \pi^0\pi^0$ decay rates are small it may be hard to carry out
this analysis. Here we show that an upper bound on the error on $\sin 2\alpha$
due to penguin diagram effects can be obtained using only the measured rate
$\BR(B^\pm \to \pi^\pm \pi^0)$ and an upper bound on the combined rate $\BR(B
\to \pi^0 \pi^0) + \BR(\bar B \to \pi^0 \pi^0)$. Since no b flavor tagging is
needed to measure this combined rate, the bound that can be achieved may be
significantly better than any approach which requires separate flavor-tagged
neutral pion information.
