Angular Size Calculator (Small-Angle Formula)

Formula

How this is calculated

The angular size θ is the angle an object subtends at your eye or telescope. The exact relationship is θ = 2·arctan(D / 2d), where D is the object's real (linear) size and d is the distance to it. Both D and d must be expressed in the same length units — the ratio D/d is dimensionless, so light-years, kilometres, or metres all work as long as you stay consistent. The result is reported in arcseconds, arcminutes, and degrees (1° = 60′ = 3600″).

Enter any two of the three quantities — actual size D, distance d, or angular size θ (in arcseconds) — and the calculator solves for the third. Given D and d it evaluates the arctangent directly; given θ and d it returns D = 2d·tan(θ/2); given θ and D it returns d = D / (2·tan(θ/2)). The displayed small-angle line uses the astronomer's shortcut θ[arcsec] ≈ 206265 · D/d, since there are 206264.8 arcseconds in one radian.

The small-angle approximation is accurate to better than 1% for angles under about 10°, which covers virtually all astronomical objects. For wide angles it slightly overestimates the angle, so the exact arctangent form is used for the hero result. Edge cases such as zero distance or a zero angle (which would make the object infinitely far) return no result.