Solar elevation + azimuth calculator

How this is calculated

The sun's position is fixed by three angles: your latitude, the sun's declination (how far north or south of the equator it stands on that date) and the hour angle (how far it has turned past solar noon, 15° per hour). The calculator works out declination and the equation of time from the day of the year, converts your clock hour to solar time, and then solves the spherical triangle between the pole, the zenith and the sun.

Declination δ = 23.44° × sin(360/365 × (N − 81)), where N is the day of year and 23.44° is the tilt of Earth's axis.
Equation of time (minutes) = 9.87 sin 2B − 7.53 cos B − 1.5 sin B, with B = 360/365 × (N − 81); this is the clock's drift from the sun, between about −14 and +16 minutes over the year.
Solar time = clock time + 4 min/° × (longitude − 15° × UTC offset in hours) + equation of time. The 4 minutes per degree is Earth's rotation; 15° × offset is the standard meridian of your time zone.
Hour angle H = 15° × (solar time − 12).
sin(elevation) = sin φ sin δ + cos φ cos δ cos H, with φ the latitude.
cos(azimuth) = (sin δ − sin(elevation) sin φ) ÷ (cos(elevation) cos φ), reflected to 360° − azimuth after solar noon so that 0° is north, 90° east, 180° south, 270° west.

Sunrise and sunset are the hour angles at which the sun's centre reaches a zenith angle of 90.833°, not 90°. Sunrise is defined as the first appearance of the sun's upper edge on a flat, sea-level horizon. The sun's apparent radius is about 0.267°, and atmospheric refraction near the horizon lifts the image by about 0.567°, so the geometric centre is 0.833° below the horizon at that moment. NOAA's solar calculator uses this same convention ("we assume 0.833° of atmospheric refraction"), so the times here are directly comparable with it. cos H₀ = (cos 90.833° − sin φ sin δ) ÷ (cos φ cos δ); sunrise and sunset are solar noon ∓ H₀ ÷ 15° in hours, converted back to clock time. If cos H₀ falls outside −1 to 1 the sun never rises or never sets that day and the result shows "—".

The time zone input is in minutes because several zones are not whole hours (India +330, Nepal +345, Newfoundland −210). It converts your clock reading to solar time; get it wrong by an hour and every output time shifts by an hour while elevation and azimuth read as if for a different hour. Arizona is −420 all year: A.R.S. 1-242 fixes the state on the solar time of the 105th meridian (Mountain Standard Time, UTC−7) and rejects daylight saving time. A clock in Arizona therefore never has to be corrected for DST, which is why Arizona sunrise tables look an hour "early" against New Mexico's in summer.

Date used is shown because the calculator never fails on a bad date. If the date field is empty or unreadable it uses today, and the result row tells you so. Latitude is clamped to ±89.9°, longitude to ±180°, hour to 0–24, offset to ±840 minutes.

Cochise County defaults

Latitude 31.3°–32.3° N, longitude −109.5° to −110.3°. Douglas 31.35° N, −109.54°; Sierra Vista 31.55°, −110.28°; Sunsites 31.94°, −109.84°; Willcox 32.24°, −109.83°. The default −109.9° is the Sulphur Springs Valley. UTC offset −420, every day of the year.

Solar noon is not 12:00. At 109.9° W the county sits 4.9° west of the 105° standard meridian, so the sun arrives about 20 minutes late, and the equation of time then moves that by up to a quarter hour either way. The calculator gives solar noon at 31.5° N, −109.9° of 12:34 on 11 February, 12:28 on 20 March, 12:21 on 21 June, 12:12 on 22 September, 12:03 on 3 November and 12:18 on 21 December. If you are using the noon shadow of a stake to find true south, use the solar-noon time from this page, not the clock's 12:00; at 12:00 on 11 February the sun is still 11° east of south.

Sun angles that matter here. On 21 December at solar noon the sun stands 34.9° above the southern horizon; a 3 m mesquite casts a 4.3 m shadow due north (height × 1.43). At 15:00 the same day it is 22.8° high at azimuth 220°, and the shadow is 2.4 times the height, reaching north-east. On 21 June at solar noon the sun is 80.7° up, almost overhead, and a south wall's shadow from a 1 m overhang is only 0.16 m long. Those three numbers decide where the array goes relative to the trees and how deep the porch roof should be.

Sunrise at 31.5° N runs from about 05:15 MST at the June solstice to 07:15 at the December solstice; sunset from 19:27 to 17:21. Daylight is 14 h 12 min in June and 10 h 6 min in December, flat-horizon times; the Dragoon, Chiricahua and Huachuca ranges take minutes off both ends depending on where you stand.

A worked example

Site at 31.5° N, 109.9° W. Date 21 June 2026, local hour 12 (noon by the clock), UTC offset −420.

This calculator: elevation 80.69°, azimuth 148.66°, sunrise 05:15, solar noon 12:21, sunset 19:27, date used 2026-06-21. The azimuth is well east of south because the clock says noon but solar noon is 21 minutes away, and near the solstice at this latitude the sun crosses almost overhead, so its bearing swings fast through midday.

The same inputs through the NOAA solar calculator's spreadsheet equations (Meeus, with the same 90.833° zenith): elevation 80.64°, azimuth 148.13°, sunrise 05:15, solar noon 12:21, sunset 19:28. The differences are 0.05° in elevation, 0.5° in azimuth and one minute in sunset. On 21 December at 12:00 the two agree to 0.01° in elevation (34.89° against 34.90°), 0.1° in azimuth (174.90° against 175.01°) and to the minute on sunrise 07:15, noon 12:18 and sunset 17:21. The residual comes from this page's single-term declination and three-term equation of time; NOAA's full series is more exact, and both ignore your horizon.

What this does not cover

Refraction in the elevation figure. The 0.833° allowance is applied to sunrise and sunset only; the elevation shown at other hours is geometric, which understates the apparent sun by up to about half a degree within a few degrees of the horizon and by nothing worth noting above 15°. Your actual horizon: every time here is for a flat, sea-level horizon, and a ridge 5° high to the east delays visible sunrise by 20–25 minutes. Elevation above sea level, which advances sunrise slightly. Clock drift in the simplified formulas for years far from 2000. The Navajo Nation or any other jurisdiction whose clock differs from the state's; the offset you enter is the one that matters. It does not model irradiance, shading losses or panel output; it tells you where the sun is, and the tilt calculator and sizing calculator take it from there.

FAQ

What UTC offset do I enter for Arizona?

−420 minutes, all year. Arizona keeps Mountain Standard Time (UTC−7) under A.R.S. 1-242 and rejects daylight saving time. California is −480 in winter and −420 in summer; New Mexico is −420 in winter and −360 in summer. If your result is off by exactly an hour, the offset is wrong.

Why does the calculator use a 90.833° zenith for sunrise and sunset?

Sunrise is when the top edge of the sun appears on a flat horizon, not when its centre is at 90° from overhead. The sun's radius is about 0.267° and atmospheric refraction lifts the image near the horizon by about 0.567°, so the centre is geometrically 0.833° below the horizon at the moment you see the first light. NOAA's calculator uses the same convention, which is why the times here match it to within a minute.

How close is this to the NOAA solar calculator?

At 31.5° N, 109.9° W on 21 June 2026 at 12:00 MST this page gives elevation 80.69° and azimuth 148.66°; the NOAA spreadsheet equations give 80.64° and 148.13°. Sunrise, solar noon and sunset agree to within one minute (05:15, 12:21, 19:27 against 05:15, 12:21, 19:28). The difference is the simplified declination and equation-of-time formulas used here; NOAA uses the full Meeus algorithms.

Sources

• NOAA Global Monitoring Laboratory, Solar Calculator — reference tool for sunrise, sunset, solar noon, azimuth and elevation (no longer actively maintained by NOAA).
• NOAA Solar Calculator, calculation details — equations from Meeus, Astronomical Algorithms; 0.833° refraction allowance; accuracy within a minute between ±72° latitude.
• A.R.S. 1-242, Standard time; exemption from daylight saving time — Arizona on the solar time of the 105th meridian, DST rejected.
• Sierra Vista, Arizona, Willcox, Arizona, Douglas, Arizona — town coordinates.
• Cochise Land Management, "Grid Power or Solar?" — Sunsites coordinates and elevation used in the county's PVGIS model.

Related guides

• Off-Grid Solar System Sizing Guide for Cochise County
• Off-Grid Solar Sizing for Cochise County Properties: A Practical Guide
• The Building Pad Spec: What to Write Down Before Anyone Quotes It