The reconstruction of images from data modeled by circular or spherical mean Radon transforms plays an important role in thermoacoustic and photoacoustic tomography. We consider a modification of a summability-type approximate reconstruction method described in earlier work and show that in the limit it leads to exact reconstruction. Among the consequences of this development are certain two- and three-dimensional inversion-type formulas in which the detectors lie on ellipses or ellipsoids respectively.