small-angle approximation
Suppose you know how big something looks in the sky (its angular size) and you also know how far away it is — can you work out its true size? For the very small angles common in astronomy, there is a wonderfully simple shortcut that turns this into a single multiplication. It is called the small-angle approximation.
When an angle is tiny, the arc it subtends and the straight chord across it are almost identical, so the messy trigonometry collapses to a clean rule: physical size is approximately the angular size times the distance, provided the angle is measured in radians. In practice astronomers use a friendly version: an object's true size equals its distance multiplied by its angular size in arcseconds, divided by about 206,265 (the number of arcseconds in a radian). Turned around, this same relation converts a measured angle into a distance — which is exactly how a one-arcsecond parallax defines a parsec.
This approximation is the quiet workhorse linking angle to true size across the field. It tells you how big a sunspot is from its apparent size, how large a galaxy is from its angular span and distance, and how far away a star is from its parallax. It is honest about its own limits: the trick only works for small angles. For something spanning many degrees, like the whole sky or a nearby planet seen up close by a spacecraft, you must use full trigonometry instead.
The Moon is about 384,000 km away and spans about 0.5 degrees (1800 arcseconds). Plugging into the small-angle rule, its true diameter comes out to roughly 3,400 km — within a few percent of the real 3,474 km, with nothing but an angle and a distance.
Angle plus distance gives true size — the small-angle approximation in action.
The approximation is only valid for small angles — it quietly fails as the angle grows. The radian, not the degree, is the natural unit that makes the simple form 'size = angle x distance' exactly true in the limit.