Angular Size & Physical Size Calculator

Give any two of angular size, physical size, and distance, and get the third — using the exact trigonometric relation, with the small-angle approximation shown alongside for comparison.

Angular / physical size calculator

Angular size is how large an object looks from here — its apparent extent on the sky. Physical size is its true linear diameter. The same physical size looks smaller the farther away it is; angular size alone never tells you which.

31.0758
Arcseconds1864.55 ″
Arcminutes31.0758 ′
Degrees0.517929 °
Radians0.00904 rad
θ = 31.0758 ′observerD (physical size)d

Schematic — the angle is exaggerated for visibility and this is not drawn to true scale (real θ ranges from arcseconds to well over the diagram's own angle). The printed θ is the actual computed value.

Exact (trigonometric)31.0758 ′
Small-angle approximation31.0757 ′
Difference0.00% — Excellent

The Moon and the Sun look almost exactly the same size in Earth’s sky — not because they’re actually similar in size (the Sun is about 400 times wider), but because the Sun also happens to be about 400 times farther away. That’s the whole idea behind angular size: how big something looks depends on both its true size and its distance, and you can’t tell which from the angle alone. This calculator goes the other way — give it any two of angular size, physical size, and distance, and it solves for the third.

The geometry

Picture an object of true diameter D, centered a distance d away. The angle it subtends, θ, is the apex angle of the isosceles triangle formed by you and the two ends of the object — which splits into two right triangles with hypotenuse d and opposite side D/2:

sin⁡ ⁣(θ2)=D/2d⟹θ=2arcsin⁡ ⁣(D2d)\sin\!\left(\frac{\theta}{2}\right) = \frac{D/2}{d} \qquad\Longrightarrow\qquad \theta = 2\arcsin\!\left(\frac{D}{2d}\right)

This is exact for any angle from 0° up to (but not including) 180°. The calculator solves this relation — or the equivalent form for D or d, whichever you’re solving for — using this trigonometric formula directly, not an approximation.

The small-angle approximation, and when it holds

For small θ, sin(θ/2) ≈ θ/2, and the exact relation collapses to the familiar rule of thumb astronomers actually use day to day:

θ≈Dd\theta \approx \frac{D}{d}

This is the version behind the classic “206265 arcseconds per radian” shortcut, and it’s genuinely excellent for almost everything in astronomy — the Moon (θ ≈ 0.5°), a galaxy cluster, a distant quasar are all comfortably in the regime where the error is a small fraction of a percent. It stops being trustworthy only for angles that are a significant fraction of a full circle — think an object close enough to fill a large part of your view, not anything actually observed through a telescope. The calculator always computes the exact value, but shows the small-angle result and the percent difference between them side by side, so you can see directly where the approximation is earning its keep and where it would mislead you — try the “very close object” preset to see it break down entirely.

Angular size vs. physical size

These are easy to conflate but describe different things. Physical size is an object’s actual linear diameter — a number in meters or light-years that doesn’t depend on who’s looking or from where. Angular size is how large it appears, and depends equally on physical size and distance — the same galaxy would look larger if it were closer, without changing at all. A measured angular size, on its own, is not a measurement of true size; you need the distance too, which is exactly what this calculator makes explicit by requiring (or solving for) all three quantities together.