  We study the space of smooth Riemannian structures on compact three-manifolds
with boundary that satisfies a critical point equation associated with a
boundary value problem, for simplicity, Miao-Tam critical metrics. We provide
an estimate to the area of the boundary of Miao-Tam critical metrics on compact
three-manifolds. In addition, we obtain a B\"ochner type formula which enables
us to show that a Miao-Tam critical metric on a compact three-manifold with
positive scalar curvature must be isometric to a geodesic ball in $\Bbb{S}^3.$
