Our understanding of RNA functions in the cell is evolving rapidly. As for proteins, the detailed three-dimensional (3D) structure of RNA is often key to understanding its function. Although crystallography and nuclear magnetic resonance (NMR) can determine the atomic coordinates of some RNA structures, many 3D structures present technical challenges that make these methods difficult to apply. The great flexibility of RNA, its charged backbone, dearth of specific surface features, and propensity for kinetic traps all conspire with its long folding time, to challenge in silico methods for physics-based folding. On the other hand, base-pairing interactions (either in runs to form helices or isolated tertiary contacts) and motifs are often available from relatively low-cost experiments or informatics analyses. We present RNABuilder, a novel code that uses internal coordinate mechanics to satisfy user-specified base pairing and steric forces under chemical constraints. The code recapitulates the topology and characteristic L-shape of tRNA and obtains an accurate noncrystallographic structure of the Tetrahymena ribozyme P4/P6 domain. The algorithm scales nearly linearly with molecule size, opening the door to the modeling of significantly larger structures.