How to Calculate the Hour Angle

Updated April 17, 2017

The determination of time, especially in daylight, has plagued man since the beginning. Without a way to measure time and keep track of it, there would be no way to establish a schedule. In ancient times, the sundial indicated the time between sunrise and sunset. It was broken down into hour-long divisions. Linear movement of the shadow cast by the sun allows you to break down the distance on the sundial between noon and sunrise or sunset evenly. The angle hour is the angle travelled by the sun in one hour. The exact value of the angle hour is dependent on your position on the Earth and the day of the year.

Look up what your latitude is. Express this latitude in terms of degrees in decimal form. Convert the minutes and seconds of angle into decimal equivalents of a degree. To convert from minutes and seconds to decimal format, divide the number of minutes by 60 and add this to the number of seconds divided by 3,600.

Determine what day of the year it is using Jan. 1 as day one. For example, April 1 is the 90th day of the year, unless it is a leap year, which would make it the 91st day.

Calculate the declination angle for today by using the following equation: D = (23.45 X pi) / 180 X sin (2 X pi X ((284 + n) / 365.25)). D represents the declination and n is the day of the year. Sin is the sine function on your calculator.

Calculate the maximum hour angle on your day, which the following formula gives: invcos ((-sin D X sin lat) / (cos D X cos lat)). Taking the inverse of the sine or cosine function is marked on calculators as arcsin/arcos or sin-1/cos-1. D in the equation represents the declination and "lat" represents your latitude.

Calculate the hour angle (HA) at sunrise and sunset by using this equation: cos HA = (-tan lat) X (tan D). The value of the hour angle is negative from sunrise until solar noon, after which the hour angle is positive. At solar noon, the value of the angle hour is zero. The hour angle changes 15 degrees for every hour.

Things You'll Need

  • Calendar
  • Calculator
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About the Author

Sean Lancaster has been a freelance writer since 2007. He has written for Writers Research Group, Alexis Writing and the Lebanon Chamber of Commerce. Lancaster holds a Doctor of Philosophy in chemistry from the University of Washington.