Rainbow Angle and Geometry Calculator

The Physics of Rainbows A rainbow is formed by sunlight refracting into a water droplet, reflecting off the back, and refracting out. Each color (wavelength) bends at a slightly different angle due to dispersion. The primary rainbow appears at about 42° from the anti-solar point (directly opposite the Sun). René Descartes (France, 1637) first explained the geometry mathematically. Snell’s law: n₁ sin(θ₁) = n₂ sin(θ₂)

Primary Rainbow (42°) One internal reflection inside the droplet. Red on outside (~42°), violet on inside (~40°), an arc of roughly 2° width. The sky inside the primary arc is brighter (Alexander’s band above it is darker). Colors: red, orange, yellow, green, blue, indigo, violet (ROYGBIV from outside to inside).

Secondary Rainbow (51°) Two internal reflections inside the droplet. Appears about 9° outside the primary rainbow, so higher in the sky. Colors run the other way: red on the inside edge (~50°), violet on the outside (~54°). That reversal is the easiest way to tell a real secondary bow from a lens flare in a photo. About 1/10 the brightness of the primary, since light escapes at every reflection.

Descartes Ray The minimum deviation angle of light through a spherical droplet, which is where the rays pile up and the bow gets its brightness. Deviation after k internal reflections is D = 2(θᵢ − θᵣ) + k(180° − 2θᵣ).

For the primary bow (k = 1): D = 180° + 2θᵢ − 4θᵣ ≈ 137.5°, so you look 180° − 137.5° = 42.5° away from the anti-solar point. For the secondary (k = 2): D = 360° + 2θᵢ − 6θᵣ ≈ 230°, and the bow sits at D − 180° ≈ 50°. θᵢ = incidence angle, θᵣ = refraction angle; using Snell’s law with n_water ≈ 1.331 to 1.343.

Viewing Geometry The anti-solar point: exactly opposite the Sun from the observer. In the morning with Sun rising in the east: look west for rainbows. At noon with Sun overhead: rainbow would be at or below the horizon (rarely visible). For maximum rainbow visibility: solar elevation angle < 42°.

Supernumerary Bows Inside the primary rainbow, faint bands of pink and green may appear. These are caused by wave interference (not explained by Descartes’ ray geometry). Visible with very uniform small droplets.