Rainbow Angle and Geometry Calculator
Calculate rainbow angles, width, and geometry.
Find the primary (42°) and secondary (51°) rainbow angles, explain the geometry, and calculate where to look for a rainbow.
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 ~2° width. The sky inside the primary arc is brighter (Alexander’s band below 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 ~9° above the primary rainbow. Colors are reversed: violet on outside (~50°), red on inside (~54°). About 1/10 the brightness of primary (more light lost at each reflection).
Descartes Ray The minimum deviation angle of light through a spherical droplet (where most light exits). For primary bow: D_min = π + 2θᵢ − 4θᵣ = 137.5° → antisolar angle = 180° − 137.5° = 42.5° θᵢ = incidence angle, θᵣ = refraction angle; using Snell’s law with n_water ≈ 1.331–1.343
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.