Systems with low mechanical dissipation are extensively used in precision measurements such as gravitational wave detection, atomic force microscopy, and quantum control of mechanical oscillators via optomechanics and electromechanics. The mechanical quality factor (Q) of these systems determines the thermomechanical force noise and the thermal decoherence rate of mechanical quantum states. While the dissipation rate is typically set by the bulk acoustic properties of the material, by exploiting dissipation dilution, mechanical Q can be engineered through geometry and increased by many orders of magnitude Recently, soft clamping in combination with strain engineering has enabled room temperature quality factors approaching 10(9) in millimeter-scale resonators. Here we demonstrate a new approach to soft clamping which exploits vibrations in the perimeter of polygon-shaped resonators tethered at their vertices. In contrast to previous approaches, which rely on cascaded elements to achieve soft clamping, perimeter modes are soft clamped due to symmetry and the boundary conditions at the polygon vertices. Perimeter modes reach Q's of 3.6 x 10(9)-a record at room temperature-while spanning only two acoustic wavelengths. We demonstrate thermal-noise-limited force sensitivity of 1.3 aN/root Hz for a 226 kHz perimeter mode with quality factor of 1.5 x 10(9) at room temperature. The small size of our devices makes them well suited for near-field integration with microcavities for quantum optomechanical experiments. Moreover, their compactness allows the realization of phononic lattices. We demonstrate a one-dimensional Su-Schrieffer-Heeger chain of high-Q perimeter modes coupled via nearest-neighbour interaction and characterize the localized edge modes.