Sun angle calculator

How is the sun's position calculated?

With the method of NOAA's solar calculator, based on Jean Meeus's Astronomical Algorithms. The date and time give the sun's declination and the equation of time; with your latitude and longitude they give the hour angle, and from that the elevation and azimuth, with a correction for the bending of light in the atmosphere. For New York City at 12:00 EDT on June 21, 2026, NOAA's own calculator and this one give an elevation of 68.9° and an azimuth of 140.5°.

How high is the sun at noon?

At solar noon the sun's elevation is 90° minus the difference between your latitude and the sun's declination: 90° − |latitude − declination|. In Washington, DC (38.9° N) on the June solstice, when the declination is about 23.4° N, that is about 74.5°; at the December solstice it is about 27.7°. Solar noon is rarely 12:00 on the clock.

How do I find the length of a shadow?

Shadow length = object height ÷ tan(sun elevation). At 45° the shadow is as long as the object is tall; at 30° it is 1.7 times as long; at 68.9° a 6-foot person casts a shadow of about 2.3 feet.

What does the azimuth mean?

The azimuth is the compass direction of the sun measured clockwise from true north: 90° is east, 180° south and 270° west. It is measured from true north, not magnetic north, so a compass needs the local magnetic declination applied.

Why is the sun's elevation slightly higher than the geometric value?

The atmosphere bends sunlight, so the sun looks higher than it is. NOAA's formula adds about 0.48° at the horizon, around 0.03° at 25° elevation, and nothing above 85°. The elevation shown here includes it, as NOAA's does.

Noon sun at the solstices across the U.S.

In 2026 the sun's declination reaches 23.4° at the June solstice and -23.4° at the December one. Applying the noon-elevation rule (90° minus the gap between latitude and declination) gives the highest and lowest the sun gets at each city, and the noon shadow of a 6-foot person:

• Miami (25.8° N): 87.7° in June (shadow 0.2 ft), 40.8° in December (shadow 7.0 ft)
• Phoenix (33.4° N): 80.0° in June (shadow 1.1 ft), 33.1° in December (shadow 9.2 ft)
• Washington, DC (38.9° N): 74.5° in June (shadow 1.7 ft), 27.7° in December (shadow 11.4 ft)
• Chicago (41.9° N): 71.6° in June (shadow 2.0 ft), 24.7° in December (shadow 13.1 ft)
• Seattle (47.6° N): 65.8° in June (shadow 2.7 ft), 19.0° in December (shadow 17.5 ft)
• Anchorage (61.2° N): 52.2° in June (shadow 4.7 ft), 5.3° in December (shadow 64.1 ft)

The rule works for places where the sun stays south of the zenith, as in the listed cities. It also shows why winters feel so different at high latitudes: in Anchorage the December noon sun is only about 5° above the horizon, so shadows are about 11 times as long as the person is tall.

Worked example: shadow of a building

A 30-foot building in Washington, DC casts a shadow 8.3 ft long at solar noon in June (74.5° elevation) and 57.2 ft long at solar noon in December (27.7°). Mornings and evenings give much longer shadows because the sun is lower, and the shadow points away from the sun, so at noon in the northern U.S. it falls toward the north. This is also the principle behind roof overhangs that shade windows in summer but let in the low winter sun.

Common mistakes

• Reading the azimuth from magnetic north. It is measured from true north, so apply your local magnetic declination before using a compass.
• Using clock time as solar time. Time zones, daylight saving time and the equation of time put solar noon anywhere from before 12:00 to well after it; the calculator takes the clock time and the time zone for the place and date.
• Dropping the sign of the longitude. It is entered in degrees east, so every place in the U.S. has a negative value (Washington, DC is about −77.04); a positive one puts the place in the eastern hemisphere.
• Using the noon rule for a different hour. It applies to solar noon only; other times need the full calculation, which the calculator does.

For the times of sunrise and sunset see the sunrise sunset calculator; to size solar panels, the solar panel calculator; to measure a tree from the angle to its top, the tree height calculator.

Sources: NOAA Global Monitoring Laboratory, Solar Calculator (gml.noaa.gov/grad/solcalc), whose algorithm is based on Jean Meeus, Astronomical Algorithms. We checked the results against the elevation, azimuth, declination and equation of time that NOAA’s calculator gives for 13 places and times. NOAA describes the results as approximate, since the atmosphere varies, and for research and recreational use only. NOAA’s own spreadsheets are valid for 1901 to 2099, the dates you can pick here. How we check our calculators.