We present results derived from the first multi-chord stellar occultations by the transneptunian object (50000) Quaoar, observed on 2011 May 4 and 2012 February 17, and from a single-chord occultation observed on 2012 October 15. If the timing of the five chords obtained in 2011 were correct, then Quaoar would possess topographic features (crater or mountain) that would be too large for a body of this mass. An alternative model consists in applying time shifts to some chords to account for possible timing errors. Satisfactory elliptical fits to the chords are then possible, yielding an equivalent radius R sub(equiv) = 555+ or -2.5 km and geometric visual albedo p sub(v) = 0.109+ or -0.007. Assuming that Quaoar is a Maclaurin spheroid with an indeterminate polar aspect angle, we derive a true oblateness of member of = 0.087 super(+0.0268) sub(-0.0175), an equatorial radius of 569 super(+24) sub(-17)km, and a density of 1.99 + or - 0.46 g cm super(-3). The orientation of our preferred solution in the plane of the sky implies that Quaoar's satellite Weywot cannot have an equatorial orbit. Finally, we detect no global atmosphere around Quaoar, considering a pressure upper limit of about 20 nbar for a pure methane atmosphere.