geometry - Why do sunbeams diverge even though the sun is much more than a few kilometers away?

Consider this picture of sun beams streaming onto the valley through the clouds.

Given that the valley is only (at a guess) 3km wide, with simple trigonometry and the angles of the beams, this gives the result that the position of the light source is being a few tens of km away at most.

What is wrong with the analysis?

Answer

This picture (source) should pretty much answer your question:

The train's destination is not above the ground, but rather far away, and perspective means that the tracks appear not to be parallel but instead to converge to the vanishing point.

The same applies to the beams of light above them. The Sun is very far away and the beams are pretty much parallel, but they're pointing towards you, and perspective makes them appear to converge towards the vanishing point - which in this case is the Sun's location in the sky.

The technical term for these beams is "crepuscular rays." Occasionally, when the Sun is very low on the horizon, you can see "anticrepuscular rays," where the beams seem to converge to a different point on the opposite side of the sky to the Sun. Here's an example: (source)

This happens for the same reason - the rays are really parallel, and there's another vanishing point in the opposite direction from the Sun.