Angular size, apparent diameter and how to compute it
Why nearby things look bigger, the exact arctangent formula, and how degrees, arcminutes and arcseconds fit together.
Apparent size versus real size
A coin held at arm's length can blot out the Moon, yet the Moon is nearly four thousand kilometres wide. That is angular size at work: what your eye registers is an angle, set by the ratio of an object's true width to its distance. Bring an object closer and that ratio grows, so it looms larger; push it away and the same object shrinks to a point. Real size never changes, but apparent size depends entirely on where you view it from.
The exact formula and the small-angle shortcut
Draw a triangle from your eye to the two edges of the object and the apparent angle sits at the vertex. That gives theta = 2 x arctan(size / (2 x distance)), the exact expression this calculator uses. For distant objects the angle is tiny and the arctangent is nearly equal to its argument, which collapses to the familiar approximation theta is about size divided by distance in radians. The shortcut is fine for a planet or a star, but for a building seen from across the street the full arctangent keeps the answer honest.
Degrees, arcminutes and arcseconds
A full circle is 360 degrees, and each degree splits into 60 arcminutes, each of which splits into 60 arcseconds. The Moon spans about 0.5178 degrees, which is close to 31 arcminutes or about 1864 arcseconds, three ways of writing the same span. Small units matter in astronomy because the interesting details are fine: Jupiter is tens of arcseconds wide, and the resolving power of a telescope is quoted in arcseconds too. This tool prints all four units so you can match whichever your source uses.
Everyday and astronomical uses
Angular size turns a measurement into something you can picture in the sky or the viewfinder. Photographers use it to judge how large a distant mountain will render at a given focal length. Astronomers use it to predict whether a galaxy fits inside an eyepiece field, or to compare a planet's apparent disc month to month as its distance changes. The same math tells a hiker that a person 1.8 metres tall a kilometre away spans only about a tenth of a degree, barely more than a distant streetlight.