Angular Size & Physical Size Calculator

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, which by the way, is a very interesting coincidence.

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.

The geometry

Picture a flat object of true diameter $D$ , centered a distance $d$ away and facing the observer broadside-on — its diameter perpendicular to the line of sight, like a disc or a ruler held up, not a ball the observer could stand inside of. The angle it subtends, $\theta$ , is the apex angle of the triangle formed by the observer and the two ends of the object. The triangle splits into two right triangles with the right angle at the object’s own center, adjacent side $d$ , and opposite side $D/2$ :

This is exact for any angle from 0° up to (but not including) 180°, for any physical size relative to the distance — a wider, closer object just pushes $\theta$ toward 180° without ever reaching it.

The small-angle approximation, and when it holds

For small $\theta$ , $\tan(\theta/2) \approx \theta/2$ (also holds for $\tan(\theta) \approx \theta$ , but naturally the approximation threshold for $\theta$ doubles), and the exact relation collapses to:

The calculator always computes the exact value, but shows the small-angle result and the percent difference between the approximation and true-size side by side, so you can see directly where the approximation is “close-enough” and where it wouldn’t be.

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 a change in its true size at all. A measured angular size, on its own, is not a measurement of true size; you need the distance too.