Which machine shop instrument is used for checking the flatness of plane surfaces to tenths of thousands of a centimeter?

The planimeter is a device specifically designed to measure the area of a two-dimensional shape, typically used in applications involving geometric measurements rather than checking the flatness of surfaces down to precise tolerances.

In the context of checking the flatness of plane surfaces, the micrometer, either in its standard form or as a blade type, is more suitable in that it can accurately measure dimensions and heights to a precise degree, including the ability to attain measurements in tenths of thousands of a centimeter. The dial caliper also offers measurement capabilities but is often less precise than a micrometer for such fine measurements.

Thus, the correct answer reflects that a micrometer, particularly a blade type micrometer, is the appropriate instrument used for assessing flatness with extreme precision—a critical factor in quality control and machine design.