Solar Angle Calculator
Calculate solar elevation (altitude above the horizon) and azimuth (compass bearing) for any location, day of year, and time of day.
Understanding Solar Angles: Elevation, Azimuth, and the Sun’s Path
The position of the sun in the sky at any moment can be precisely described by two angles: solar elevation and solar azimuth. These angles depend on three factors — where you are on Earth (latitude), what time of year it is (day of year), and what time of day it is (hour). Knowing these angles is fundamental to solar energy engineering, architecture, photography, and navigation.
Solar Elevation (Altitude) Angle
Solar elevation — also called the altitude angle — measures how high the sun is above the horizon, in degrees. An elevation of 0° means the sun is right at the horizon (sunrise or sunset). An elevation of 90° means the sun is directly overhead (the zenith), which only occurs in the tropics on specific dates.
At higher latitudes, the maximum possible elevation angle is lower. For example, at 51° N latitude (London), the maximum midday elevation is about 62° in summer and just 15° in winter. Lower elevation angles mean sunlight travels through more atmosphere, scattering more energy — which is why winter days feel less intense even when skies are clear.
The elevation angle is calculated using the formula: El = arcsin(sin(φ)·sin(δ) + cos(φ)·cos(δ)·cos(HA)), where φ is latitude, δ is solar declination, and HA is the hour angle.
Solar Azimuth Angle
Solar azimuth describes the compass direction of the sun — measured clockwise from true north. At azimuth 0°, the sun is due north; at 90°, due east; at 180°, due south; at 270°, due west.
In the Northern Hemisphere, the sun rises in the east, reaches its peak in the south at solar noon, and sets in the west. The exact azimuth at sunrise and sunset changes throughout the year: at the summer solstice, the sun rises well north of east and sets well north of west; at the winter solstice, it rises and sets south of east and west respectively.
In the Southern Hemisphere, the sun’s midday position is in the north, and the azimuth pattern is mirrored accordingly.
Solar Declination
Solar declination (δ) is the angle between the Earth’s equatorial plane and the line connecting Earth’s center to the Sun. It varies from approximately −23.45° at the December solstice to +23.45° at the June solstice.
This seasonal variation is caused by Earth’s axial tilt of about 23.5°. When the Northern Hemisphere tilts toward the sun (June), declination is positive and northern latitudes receive more direct sunlight. When it tilts away (December), declination is negative and days are shorter and less intense in the north.
The declination formula used here is: δ = 23.45° × sin(360/365 × (284 + N)), where N is the day of the year. This approximation is accurate to within 0.3° for most practical purposes.
Hour Angle and Solar Noon
The hour angle (HA) measures how far the sun has moved from the meridian (solar noon). Earth rotates 360° in 24 hours, or 15° per hour. Therefore, HA = 15° × (hour − 12). At solar noon, HA = 0°. One hour before noon, HA = −15°; one hour after noon, HA = +15°.
Note that solar noon does not coincide exactly with clock noon. Solar time depends on your true longitude within your time zone, and can differ from clock time by up to an hour or more. This calculator uses solar time directly (hour input = solar time), so results reflect the sun’s true geometric position.
At solar noon, the sun achieves its maximum elevation for the day. The midday elevation equals 90° − |φ − δ| for locations where the sun crosses due south (or north in the Southern Hemisphere).
Applications
Solar angle calculations are essential for solar panel installation. Panels produce the most energy when tilted to be perpendicular to incoming sunlight. Knowing the elevation angle at different times and seasons allows engineers to choose the optimal tilt angle for a fixed installation or design a tracking system.
Architects use solar angles to design buildings that maximize passive solar gain in winter while providing shade in summer. South-facing windows (in the Northern Hemisphere) admit low-angle winter sun and can be shaded by overhangs from high-angle summer sun.
Photographers and filmmakers use solar elevation to predict golden hour — the period shortly after sunrise and before sunset when the sun is near 6° elevation, producing warm, soft light with long shadows. Azimuth data helps position subjects with the sun at the ideal angle.
Limitations of This Calculator
This calculator uses simplified astronomical formulas that treat the Earth’s orbit as circular and ignore several small corrections applied in professional-grade solar position algorithms. The results are generally accurate to within 1–2° for everyday purposes.
The hour input represents idealized solar time, not local clock time. To convert clock time to solar time, you would need to account for the equation of time and your longitude offset from the center of your time zone. For precise solar position data, specialized software or almanacs should be consulted.
Frequently Asked Questions
What is solar elevation angle?
Solar elevation angle (also called solar altitude) measures how high the sun is above the horizon in degrees. An angle of 0° means the sun is at the horizon (sunrise or sunset), while 90° means the sun is directly overhead. The elevation changes throughout the day (reaching its peak at solar noon) and throughout the year as the sun’s declination shifts with the seasons.
What is solar azimuth?
Solar azimuth is the compass direction of the sun, measured clockwise from true north. North = 0°, East = 90°, South = 180°, West = 270°. In the Northern Hemisphere, the sun is generally in the south during midday; in the Southern Hemisphere, it is in the north. The azimuth at sunrise and sunset shifts north and south with the seasons.
What is solar declination?
Solar declination is the angle between the Earth’s equatorial plane and the direction of the Sun. It ranges from −23.45° (around December 21) to +23.45° (around June 21). Declination is zero at the spring and autumn equinoxes. This seasonal variation is caused by Earth’s axial tilt and is the fundamental reason we experience seasons.
What is the hour angle?
The hour angle measures how far the sun has moved from the meridian. Earth rotates 15° per hour, so the hour angle is 15° × (solar hour − 12). At solar noon the hour angle is 0°. Negative values indicate morning (sun east of meridian) and positive values indicate afternoon (sun west of meridian).
Why does the calculator use solar time rather than clock time?
Solar time is based purely on the sun’s geometric position relative to your meridian. Clock time is offset from solar time by two factors: your position within your time zone (each degree of longitude = 4 minutes), and the equation of time (a variation of up to ±16 minutes caused by Earth’s elliptical orbit and axial tilt). This calculator uses solar time directly so the formulas are straightforward. For precise real-world results, convert your local clock time to solar time first.
When is the sun highest in the sky?
The sun reaches its highest elevation at solar noon (hour angle = 0°). The maximum elevation that day equals 90° minus the absolute difference between your latitude and the solar declination. For example, at latitude 40° N on the summer solstice (δ ≈ +23.45°), the noon elevation is approximately 90° − (40° − 23.45°) = 73.45°. On the winter solstice (δ ≈ −23.45°), it would be 90° − (40° + 23.45°) = 26.55°.