1. A wear amount measuring device comprising: an image display unit configured to: display a real object image based on real object image data containing a wear amount measurement target and a reference part thereof functioning as a measurement-related positioning reference for the wear amount measurement target; and display a plan image based on design plan data containing the wear amount measurement target and the reference part thereof; an image processing unit configured to execute an image processing of overlapping the real object image and the plan image at an equal scale on a corresponding positional relation when the reference part in the real object image and the reference part in the plan image are matched; and a wear amount computing unit configured to compute a wear amount based on a magnitude of an interval between a measurement contour line drawn along a contour of the wear amount measurement target in the real object image and a plan contour line in the plan image.