How are sunrise and sunset calculated?

From the sun's declination and your latitude. The hour angle at sunrise satisfies cos H = (cos z − sin φ sin δ) ÷ (cos φ cos δ), where φ is latitude, δ is the sun's declination for that date and z is the zenith angle at sunrise. Solar noon is found from your longitude and the equation of time, and sunrise and sunset sit symmetrically either side of it at four minutes per degree of hour angle.

Why is day longer than night at the equinox?

Two reasons, both about the definition of sunrise. It is timed to the sun's upper edge rather than its centre, which is worth about 0.27 degrees, and the atmosphere bends light so the sun appears above the horizon while still geometrically below it, worth another 0.57. Together they push sunrise earlier and sunset later by several minutes each. At the equator that gives about twelve hours and seven minutes of daylight rather than exactly twelve.

What is solar noon and why is it not 12:00?

Solar noon is when the sun is genuinely at its highest, which rarely lines up with clock noon. Two things push them apart. Your longitude within the time zone moves it by four minutes per degree from the zone's meridian, and the equation of time — caused by Earth's elliptical orbit and its axial tilt — shifts it by anything from fourteen minutes early to sixteen minutes late depending on the date. Daylight saving adds another hour on top.

What is the equation of time?

The difference between the sun's actual position and where a perfectly regular clock would put it. Earth's orbit is slightly elliptical so it moves faster in January than July, and its axis is tilted, which changes how that motion projects onto the celestial equator. The two effects combine into a curve that reaches about sixteen and a half minutes in early November and about fourteen minutes the other way in mid February, crossing zero four times a year.

What are the three kinds of twilight?

They are defined by how far the sun has dropped below the horizon. Civil twilight runs to 6 degrees below and is bright enough to read outdoors or drive without headlights. Nautical twilight runs to 12 degrees, historically the point at which sailors could still make out the horizon against the sky to take star sightings. Astronomical twilight ends at 18 degrees, after which the sky is dark enough that faint objects are no longer washed out.

What causes the midnight sun and polar night?

Earth's axis is tilted by about 23.4 degrees, so above the Arctic circle at 66.6 degrees north the sun can stay above the horizon for a full day around the June solstice, and below it for a full day around December. Mathematically the sunrise equation simply has no solution — the required cosine falls outside the range −1 to 1, meaning the sun never reaches that height. This calculator reports that directly rather than inventing a time.

Why do the earliest sunset and shortest day fall on different dates?

Because the equation of time is still shifting while day length is near its minimum. In the northern hemisphere the earliest sunset comes about two weeks before the December solstice and the latest sunrise about two weeks after, even though the shortest day is the solstice itself. Solar noon is drifting later through that period, which pulls sunrise and sunset in the same direction while the gap between them is at its narrowest.

How accurate are these times?

Within about a minute for most locations and dates, which is the practical limit of the standard algorithm. The main uncertainty is not the astronomy but the atmosphere: the standard refraction figure assumes typical temperature and pressure, and unusual conditions can shift sunrise by a minute or more. Local horizon matters more still — hills, buildings or a high vantage point will change the observed time far more than the calculation's own error.

Do I need to adjust for daylight saving?

Yes. This calculator takes a UTC offset and applies it exactly as given, so set it to match the date you are asking about rather than today. London is UTC+1 from late March to late October and UTC+0 for the rest of the year, so a June query needs +1 and a December query needs 0. Getting it wrong shifts every time by a whole hour while leaving the day length correct.

Sunrise and Sunset Calculator

Find sunrise, sunset, solar noon and day length for any latitude, longitude and date, along with the three stages of twilight and the sun's height through the whole day. Pick a city or type in coordinates, and the chart on the right traces the sun's path from midnight to midnight so you can watch it...

WHERE AND WHEN

LATITUDE

north positive, −90 to 90

LONGITUDE

east positive, −180 to 180

DATE

UTC OFFSET

set to match the date — daylight saving is not applied

CITIES

SUNRISE

04:43

SUNSET

21:22

DAY LENGTH

16h 38m

SOLAR NOON

13:02

DECLINATION

+23.44°

TWILIGHT

STAGE

BEGINS

ENDS

Astronomical

sun 18° below

—

Nautical

sun 12° below

02:41

23:24

Civil

sun 6° below

03:55

22:09

Daylight

sun above the horizon

04:43

21:22

Equation of time today: -1.82 minutes — the gap between clock noon and the sun's actual highest point.

WATCH THE DAY PASS

Sunrise is not when the sun's centre reaches the horizon. It is when the upper edge first appears, and the atmosphere bends light enough to lift the whole disc into view before it has geometrically arrived. Together those add about 0.83°, which is why day is a few minutes longer than night even at the equinox.

The swing between summer and winter grows with latitude, and the curve is steepest at the equinoxes rather than the solstices.

Created with❤️byeaglecalculator.com

HOW TO USE

1
Choose a city, or type in a latitude and longitude directly. North and east are positive, so London is about 51.51 and −0.13, while Sydney is −33.87 and 151.21.
2
Set the date and the UTC offset. The offset matters — daylight saving is not applied for you, so use +1 for British Summer Time in July but 0 for December.
3
Read the sunrise and sunset times, along with solar noon, day length and the sun's declination for that date. The twilight table below gives the start and end of all three stages.
4
Watch the chart. The sun moves along its path for the whole day, the shaded bands beneath the horizon are the twilight stages, and you can drag the slider to any moment to read the sun's exact height.

THE FORMULAS

Hour angle at sunrisecos H = (cos z − sin φ sin δ) / (cos φ cos δ)

Sunrise zenithz = 90.833°

Solar noon (UTC min)720 − 4·longitude − E

Sunrise / sunsetnoon ∓ 4H minutes

Day length8H / 60 hours

Solar elevationsin a = sin φ sin δ + cos φ cos δ cos H

Civil twilightz = 96°

Nautical / astronomicalz = 102° / 108°

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Last updated: July 29, 2026 · NOAA solar algorithm · Eagle-eyed accuracy for every calculation.