Solar Altitude and Azimuth: Meaning, Formulas & Worked Example

Direct answer

Definitions of solar altitude, azimuth, and solar noon, and how they change with season and latitude — with computed noon-altitude values for the solstices and equinox. A key reference is altitude: 0-90 degrees; check the cited source, edition, units, and applicability before using it for design, compliance, procurement, or approval. Altitude is 0-90 degrees; sun height above horizon. Azimuth is 0-360 degrees; compass bearing of the sun. Solar noon is Sun due south (N. hemisphere); daily peak altitude. Every sun position is defined by two angles measured from where you stand: Read the values with the cited source, edition or observation period, units and applicability; replace illustrative inputs with current project evidence before design, procurement or approval use. Retain the source file, retrieval date, assumptions and named checker in the project record so the result can be reproduced and reviewed when the governing information changes.

Content and cited inputs reviewed by Harth team · Published 2026-08-14 · Updated 2026-08-14.

What does this term mean?

Solar Altitude and Azimuth Explained: Definitions of solar altitude, azimuth, and solar noon, and how they change with season and latitude — with computed noon-altitude values for the solstices and equinox.

What are the key takeaways?

• Altitude: 0-90 degrees
• Azimuth: 0-360 degrees
• Solar noon: Sun due south (N. hemisphere)
• Noon altitude formula: 90 - |lat - declination|

Which solar conditions control the result?

Every sun position is defined by two angles measured from where you stand:

• Solar altitude is the angle of the sun above the horizon. It is 0 degrees at sunrise and sunset and reaches its daily maximum at solar noon.
• Solar azimuth is the sun's horizontal bearing, measured as a compass direction. It tells you which facade the sun is facing at any moment.

Solar noon

Solar noon is when the sun crosses the meridian and hits its highest point of the day — due south in the northern hemisphere. Because it is the daily peak, the noon altitude is the single most useful number for shading design. At solar noon the altitude is:

$$ \text{noon altitude} = 90 - |,\text{latitude} - \text{declination},| $$

where solar declination is about +23.44 degrees at the summer solstice, 0 degrees at the equinoxes, and -23.44 degrees at the winter solstice.

Solar-noon altitude by latitude (computed)

The table below applies the formula above. Values are approximate — they use mean declination and ignore atmospheric refraction.

Reading down a column shows how the sun sinks lower as you move toward the poles. Reading across a row shows the seasonal swing — always about 46.9 degrees between the two solstices, twice the axial tilt.

Altitude, elevation, zenith and azimuth: which is which

Before comparing numbers from two tools, check which azimuth reference each uses: a south-referenced azimuth of 0 degrees is due south, which a north-referenced tool reports as 180 degrees.

Solar altitude and azimuth formulas

These are the standard spherical-geometry relations, as published in NOAA's solar position calculation notes and PVEducation's pages on elevation, azimuth and declination. φ is latitude (positive north), n is the day of the year (1 January = 1).

Declination (Cooper's approximation):

$$ \delta \approx 23.45 \times \sin\left[\frac{360}{365},(284 + n)\right] $$

The 23.45 coefficient is Earth's axial tilt — the reason the sun's overhead latitude swings between the tropics. NASA's Earth fact sheet currently gives the obliquity as 23.44 degrees, the value this page's tables use; the 0.01 degree difference is immaterial for design. NOAA's own calculator uses a longer Fourier series for declination.

Hour angle:

$$ h = 15 \times (\text{solar time in hours} - 12) $$

Solar altitude angle:

$$ \sin\alpha = \sin\varphi,\sin\delta + \cos\varphi,\cos\delta,\cos h $$

NOAA writes the same equation for the zenith angle, since cos(zenith) = sin(altitude). At solar noon h = 0 and it reduces to the noon formula above.

Solar azimuth (clockwise from north):

$$ \cos A = \frac{\sin\delta,\cos\varphi - \cos\delta,\sin\varphi,\cos h}{\cos\alpha} $$

Arccos only returns 0–180 degrees, so use A as calculated in the morning (negative h) and 360 - A in the afternoon (positive h).

Solar time is not clock time. NOAA converts with a time offset in minutes of eqtime + 4 × longitude - 60 × timezone (longitude positive east, timezone in hours from UTC).

Worked example: 40 degrees N, 21 June, 3:00 pm solar time

Inputs: latitude φ = 40 degrees N, day n = 172 (21 June, non-leap year), solar time 15:00.

At 3:00 pm solar time the sun is about 49 degrees up and about 10 degrees south of due west — the angle a west facade and its shading must deal with on the longest day. At 9:00 am (h = -45 degrees) the altitude is the same and the azimuth mirrors it at 99.8 degrees. The noon check, 90 - |40 - 23.45| = 73.45 degrees, matches the 40 degree row of the table above.

Sun altitude and azimuth at 40 degrees N through the year (computed)

Computed with the formulas above for a site at 40 degrees N, using declinations of +23.44, 0 and -23.44 degrees. Each cell is altitude / azimuth.

Sunrise azimuths are geometric (sun's centre on a flat horizon, no refraction); NOAA times sunrise at a zenith of 90.833 degrees to allow for refraction and the solar disc, so observed sunrise comes slightly earlier. Sunset azimuths mirror sunrise about due south (for example 301.3 degrees in June).

Why this matters in design

• A high summer noon altitude means a modest horizontal overhang can shade south glazing at the hottest time of year.
• A low winter noon altitude means that same overhang still admits low-angle winter sun for passive heating and daylight.
• Low-latitude sites face near-overhead summer sun (little help from vertical shading) but strong east/west low-angle exposure; high-latitude sites deal with low sun year-round and long shadows.

For a full worked example, see sun path and solar angles for Phoenix, AZ, and read understanding sun-path diagrams for how these angles are plotted.

Solar Altitude and Azimuth: Meaning, Formulas & Worked Example: reproducibility record

Project-use verification: retain the inputs and method with every plotted result. A diagram without its coordinates, time basis and algorithm is illustrative only.

Where can you explore related guidance?

• Explore all solar data guidance
• Understanding Sun-Path Diagrams
• Sun Path & Solar Angles for Phoenix, AZ

Sources and method

NOAA Solar Calculator / standard solar geometry equations. Noon altitude computed as 90 - |latitude - declination|, with declination taken as +23.44 (summer solstice), 0 (equinox), and -23.44 (winter solstice). Values are approximate and ignore atmospheric refraction and the equation of time. Official references: NOAA solar calculation details. Altitude, azimuth and declination formulas: PVEducation (elevation, azimuth, declination); axial tilt 23.44 degrees: NASA Earth fact sheet. Worked example and 40 degrees N table computed from these equations (geometric, no refraction).

Source links

• gml.noaa.gov

• pveducation.org

• pveducation.org

• pveducation.org

• nssdc.gsfc.nasa.gov

• gml.noaa.gov