The structure of $\mathrm{Ge}(105)\ensuremath{-}(1\ifmmode\times\else\texttimes\fi{}2)$ grown on Si(105) is examined by scanning tunneling microscopy (STM) and first-principles calculations. The morphology evolution with an increasing amount of Ge deposited documents the existence of a tensile surface strain in Si(105) and its relaxation with increasing coverage of Ge. A detailed analysis of high-resolution STM images and first-principles calculations produce a new stable model for the $\mathrm{Ge}(105)\ensuremath{-}(1\ifmmode\times\else\texttimes\fi{}2)$ structure formed on the Si(105) surface that includes the existence of surface strain. It corrects the model developed from early observations of the facets of ``hut'' clusters grown on Si(001).