Angular Size Converter
Small-angle astronomy: how big objects look on the sky from Earth.
why D/d is never exactly right 🖖
The small-angle formula θ ≈ D/d is a first-order approximation of the exact relation θ = 2 arctan(D/2d). The exact formula comes from the geometry: the observer sees a chord of length D at distance d, and the full angle is twice the arctan of the half-chord over the distance. The approximation works because arctan(x) ≈ x for small x. For the Moon (D ≈ 3475 km, d ≈ 384,000 km), the ratio D/d ≈ 0.0090, and the error is just 0.002%. At 10° the error is already 0.25%; at 60° it reaches 15%. In astronomy nearly every target satisfies D ≪ d, so the approximation holds to parts per million — but for objects at close range (a football field seen from 50 m, a planet from a close flyby) the arctan formula matters.
It's about how big, not how far 🖖
Angular size measures how large something looks, not its true size — an angle, not a length. A far-off skyscraper and a coin held at arm's length can appear identical because size and distance trade off in the ratio D/d. The Sun is about 400 times wider than the Moon but also about 400 times farther away, so both span roughly 0.5° in our sky — the coincidence that makes total solar eclipses possible.
In an expanding universe, farther looks bigger 🖖
The small-angle idea assumes flat, static space, but cosmology breaks it. Because the universe expanded while ancient light traveled to us, the angular diameter distance is not monotonic: it peaks around redshift z ≈ 1.6. A galaxy of fixed size therefore shrinks on the sky only out to that point — beyond it, more distant (younger) galaxies actually appear larger, not smaller.