indirect measurement

Indirect measurement is finding a length you cannot reach with a ruler by measuring something you can, and then using a proportion to bridge the two. You cannot lay a tape measure up the side of a tree, across a canyon, or out to the Moon — but you can set up similar triangles, measure their accessible parts, and let proportion do the rest. It is geometry put to work as a measuring instrument.

The classic method uses shadows or sightlines to create two similar triangles, then solves a single proportion. To find a tree's height, stand a metre stick upright and measure its shadow and the tree's shadow at the same moment; the sun's rays make the triangle of stick-and-shadow similar to the triangle of tree-and-shadow, so (stick height)/(stick shadow) = (tree height)/(tree shadow). A mirror on the ground works the same way: line up your eye with the treetop in the mirror and the reflection makes two similar triangles. Every such method comes down to: build similar triangles, measure the parts you can reach, write the proportion, solve.

This is how surveyors, sailors, and astronomers have measured the unmeasurable for millennia — the legendary measurement of the pyramid's height by Thales and Eratosthenes' estimate of the Earth's circumference are early triumphs. The honest caveat is that the answer is only as good as your measurements and your assumption of similarity: the stick must be vertical, the ground level, and the readings taken at the same instant, or the proportion you trust will quietly mislead you.