the linear-intercept method
The most-used way to size grains is beautifully simple, like measuring how far apart fence posts are by walking a straight path and counting how many fences you cross. In the linear-intercept method you draw one or more straight test lines of known total length across a micrograph of a polished, etched section, and count how many grain boundaries those lines cut through. Divide the true line length (the drawn length divided by the magnification) by the number of boundaries crossed, and you get the mean linear intercept — the average distance the line travels inside one grain before hitting the next boundary.
As a worked example, suppose you draw test lines totalling 500 mm on an image taken at 500 times magnification, so their real length on the sample is 500 divided by 500, that is 1 mm, i.e. 1000 microns. If those lines cross 200 grain boundaries, the mean linear intercept is 1000 divided by 200, which is 5 microns. That intercept is not quite the true grain diameter, because a random line rarely crosses a grain at its widest; for space-filling equiaxed grains the accepted conversion multiplies the mean intercept by about 1.56 to estimate the true average grain diameter, so a 5 micron intercept implies grains near 8 microns. The method is codified in the standard ASTM E112, which also lets you convert the intercept count straight to an ASTM grain-size number.
It is popular because it is fast, needs no fancy software, is statistically sound, and handles irregular grains gracefully — you are counting crossings, not judging where each grain begins and ends. The keys to a trustworthy result are counting enough boundaries (several hundred, from lines placed randomly and in several directions so an elongated structure is not biased), and etching the section well enough that every boundary is actually visible, since a missed boundary inflates the apparent grain size. For elongated or textured grains, lines run in different directions reveal the anisotropy that a single direction would hide.
To certify a fine alumina, an analyst overlays five randomly-oriented test lines on an SEM image, tallies 312 boundary crossings over a true line length of 936 microns, and computes a mean intercept of 3.0 microns. Multiplying by 1.56 gives a reported grain size of 4.7 microns — repeatable by any other lab following the same standard.
Count the crossings, divide, and multiply by the shape factor — a whole microstructure reduced to one comparable number.
The 1.56 factor and the intercept-to-grain-size conversion assume equiaxed, roughly uniform grains. Apply it blindly to elongated, textured, or wildly bimodal structures and the single number it yields will hide the very features that matter most.