We measure the acoustic scale from the angular power spectra of the Sloan Digital Sky Survey III (SDSS-III) Data Release 8 imaging catalog that includes 872, 921 galaxies over ~10,000 deg2 between 0.45 < z < 0.65. The extensive spectroscopic training set of the Baryon Oscillation Spectroscopic Survey luminous galaxies allows precise estimates of the true redshift distributions of galaxies in our imaging catalog. Utilizing the redshift distribution information, we build templates and fit to the power spectra of the data, which are measured in our companion paper, to derive the location of Baryon acoustic oscillations (BAOs) while marginalizing over many free parameters to exclude nearly all of the non-BAO signal. We derive the ratio of the angular diameter distance to the sound horizon scale DA (z)/rs = 9.212+0.416 - 0.404 at z = 0.54, and therefore DA (z) = 1411 ± 65 Mpc at z = 0.54; the result is fairly independent of assumptions on the underlying cosmology. Our measurement of angular diameter distance DA (z) is 1.4sigma higher than what is expected for the concordance LambdaCDM, in accordance to the trend of other spectroscopic BAO measurements for z >~ 0.35. We report constraints on cosmological parameters from our measurement in combination with the WMAP7 data and the previous spectroscopic BAO measurements of SDSS and WiggleZ. We refer to our companion papers (Ho et al. de Putter et al.) for investigations on information of the full power spectrum.