Given an approximately centered image of a spiral galaxy, we describe an entirely automated method that finds, centers, and sizes the galaxy (possibly masking nearby stars and other objects if necessary in order to isolate the galaxy itself) and then automatically extracts structural information about the spiral arms. For each arm segment found, we list the pixels in that segment, allowing image analysis on a per-arm-segment basis. We also perform a least-squares fit of a logarithmic spiral arc to the pixels in that segment, giving per-arc parameters, such as the pitch angle, arm segment length, location, etc. The algorithm takes about one minute per galaxies, and can easily be scaled using parallelism. We have run it on all ∼644,000 Sloan objects that are larger than 40 pixels across and classified as "galaxies." We find a very good correlation between our quantitative description of a spiral structure and the qualitative description provided by Galaxy Zoo humans. Our objective, quantitative measures of structure demonstrate the difficulty in defining exactly what constitutes a spiral "arm," leading us to prefer the term "arm segment." We find that pitch angle often varies significantly segment-to-segment in a single spiral galaxy, making it difficult to define the pitch angle for a single galaxy. We demonstrate how our new database of arm segments can be queried to find galaxies satisfying specific quantitative visual criteria. For example, even though our code does not explicitly find rings, a good surrogate is to look for galaxies having one long, low-pitch-angle arm - which is how our code views ring galaxies. SpArcFiRe is available at http://sparcfire.ics.uci.edu. © 2014. The American Astronomical Society. All rights reserved.