Angular Diameter (Small-Angle)

Hold the geometry of a distant sphere at arm's length: its true diameter 2R, seen from distance d, spans an angle θ = 2R/d radians — provided the angle is small, since the exact relation is θ = 2 arctan(R/d) and the two agree to better than a part in ten thousand below about 1°, drifting to a 1 % error only past 10° or so. Astronomy lives comfortably inside that limit. The Sun: θ = 2 × 6.957 × 10⁸ / 1.496 × 10¹¹ = 9.30 × 10⁻³ rad = 0.533°. The Moon: 2 × 1.737 × 10⁶ / 3.844 × 10⁸ = 9.04 × 10⁻³ rad. Those two numbers agreeing within 3 % is the great coincidence of our sky — the Moon's disc almost exactly covers the Sun's, giving Earth total solar eclipses complete with visible corona, a spectacle no other planet in the solar system gets.

The formula is the working currency of observational astronomy in both directions. Forward: Jupiter at 4.2 au subtends 47″, which is why a modest telescope shows its cloud bands. Backward, it converts a measured angle into a true size once a distance is known — and it explains why stars stubbornly refuse to show discs: shrink the Sun to a typical stellar distance and d = 2R/θ says it would subtend one arcsecond only at about 1900 au, so even nearby giants span a few hundredths of an arcsecond, resolvable only by interferometry. When the ratio runs the other way and the angle is generous, the same line of reasoning becomes the surveyor's stadia method and the rule that a thumb at arm's length covers about 2°.