Lightly etched single-walled carbon nanotubes are chemically reacted to form rings. The rings appear to be fully closed as opposed to open coils, as ring-opening reactions did not change the structure of the observed rings. The average diameter of the rings was 540 nanometers with a narrow size distribution. The nanotubes in solution were modeled as wormlike polymer chains, yielding a persistence length of 800 nanometers. Nanotubes shorter than this length behave stiffly and stay nearly straight in solution. However, nanotubes longer than the Kuhn segment length of 1600 nanometers undergo considerable thermal fluctuation, suggesting a greater flexibility of these materials than is generally assumed.