Rainbow and Halo Angle Calculator (Why 42° and 22°)

What this tool does

A raindrop bends light on the way in, reflects it once inside and bends it again on the way out, and the light piles up at one angle where the total deviation is smallest — about 42° from the point opposite the Sun for red. Violet light bends more, so its bow sits a little lower, which is why red is on the outside of the primary. Two reflections give the fainter secondary bow at 50–54° with the colours reversed, and the dark band between the bows is called Alexander's. The 22° halo is the same idea in ice prisms.

Formula

cos² i = (n² − 1) ÷ 3 · sin r = sin i ÷ n · primary = 4r − 2i · secondary = 180° + 2i − 6r · halo = 2 asin(n/2) − 60°

Variables

Worked example

Limitations

• The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
• The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.