Daylight Duration Calculator Formula

Understand the math behind the daylight duration calculator. Each variable explained with a worked example.

Use the Daylight Duration Calculator

Formulas Used

Daylight Hours

day_hours = (2 / 15) * (180 / pi) * acos(-tan(latitude * pi / 180) * tan(declination * pi / 180))

Night Hours

night_hours = 24 - (2 / 15) * (180 / pi) * acos(-tan(latitude * pi / 180) * tan(declination * pi / 180))

Variables

VariableDescriptionDefault
latitudeLatitude(°)40.7
declinationSolar Declination(°)23.44

How It Works

Daylight Duration

The number of hours between sunrise and sunset depends on latitude and the Sun's declination (which changes through the year).

Formula

D = (2/15) × arccos(-tan(φ) × tan(δ)) hours

where φ = latitude and δ = solar declination (both in degrees, converted to radians for calculation).

Solar declination ranges from -23.44° (Dec solstice) to +23.44° (Jun solstice).

Worked Example

New York (40.7°N) at summer solstice (declination = 23.44°).

latitude = 40.7declination = 23.44
  1. 01cos(H) = -tan(40.7°) × tan(23.44°)
  2. 02= -0.860 × 0.433 = -0.3726
  3. 03H = arccos(-0.3726) = 111.86°
  4. 04D = 2 × 111.86 / 15 = 14.91 hours of daylight

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