Venusian plains regions, as imaged by the Magellan spacecraft, display many styles of tectonic and volcanic deformation. Radar images of several areas of the volcanic plains reveal polygonal patterns of bright lineations. Intersection geometries of the lineations defining the polygonal patterns are typical of those found in tensile networks. In addition, the polygonal patterns generally exhibit no preferred orientation, implying that they are the result of horizontally isotropic stress fields. Such stress fields usually arise on the Earth as a consequence of desiccation, freeze-thaw cycles, or cooling and produce mud cracks, ice-wedge polygons, and columnar joints, respectively. We propose that the polygonal patterns seen in the Magellan images of some of the volcanic plains are the result of thermal stresses. We consider two alternative scenarios which would generate sufficient tensile thermal stresses to cause failure. The first scenario is that of a cooling lava flow; the residual thermal stress which would develop (assuming no failure of the rock) is tensional and of the order of 400 MPa. This is much greater than the strength of unfractured terrestrial basalt (∼10 MPa), so we can expect joints to form during cooling of Venusian lava flows. However, the spacing of the polygonal lineations seen in Magellan images is typically 1–2 km, much larger than the largest spacings of decimeters for joints in terrestrial lavas. The second scenario involves an increased heat flux to the base of the lithosphere; the resulting thermal stresses cause the upper lithosphere to be in tension and the lower lithosphere to be in compression. Brittle tensile failure occurs near the surface due to the finite yield strength of the lithosphere. The maximum depth to which failure occurs increases with increasing elevation of the temperature gradient. For an initially 25-km-thick lithosphere and temperature gradient of 11°/km, this maximum depth varies from 0.5 km to 2 km as the temperature gradient is increased to 12°/km and 22°/km, respectively. Both the cooling flow scenario and the heated lithosphere scenario produce isotropic tensile surface stress patterns, but the heated lithosphere model is more compatible with the kilometer scale of the polygonal patterns seen in Magellan images.