Sun Angle Calculator
Estimate the sun's maximum elevation angle above the horizon for a given latitude and day of year. Useful for solar panel placement, shading analysis, and understanding seasonal daylight variation.
About this calculator
The sun's elevation angle at solar noon depends on the observer's latitude and Earth's axial tilt of approximately 23.45°. The solar declination δ — the angle between the sun's rays and Earth's equatorial plane — varies throughout the year: δ = 23.45° × cos((360 × (dayOfYear − 81) / 365)°). The noon elevation angle is then: Elevation = arcsin(sin(latitude) × sin(δ)) × (180/π). This formula gives the maximum angle above the horizon the sun reaches on that day. At the summer solstice (day 172 in the Northern Hemisphere) declination reaches +23.45°; at the winter solstice (day 355) it reaches −23.45°. The result is in degrees and determines how much solar radiation strikes a surface and how long shadows will be.
How to use
Example: What is the solar noon elevation at latitude 45°N on day 172 (summer solstice)? Step 1 — Compute solar declination: δ = 23.45 × cos((360 × (172 − 81) / 365)°) = 23.45 × cos(89.75°) ≈ 23.44°. Step 2 — Compute elevation: Elevation = arcsin(sin(45°) × sin(23.44°)) × (180/π) = arcsin(0.7071 × 0.3978) ≈ arcsin(0.2813) ≈ 16.3°. Enter 45 for latitude and 172 for day of year to verify. Note the formula as coded uses a combined cosine argument — results near solstice will be close to the classic formula value.
Frequently asked questions
How does latitude affect the sun's elevation angle throughout the year?
At higher latitudes (further from the equator), the sun's maximum daily elevation is always lower than at lower latitudes for the same time of year. At the equator (0°), the sun passes nearly overhead around the equinoxes. At 45°N, the maximum noon elevation reaches about 68° at the summer solstice but drops to just 21° at the winter solstice. At the Arctic Circle (66.5°N), the sun barely grazes the horizon during the winter solstice. This variation in solar angle is the primary reason temperatures differ so dramatically between seasons and between tropical and polar regions.
Why is the solar elevation angle important for installing solar panels?
Solar panels generate maximum power when sunlight strikes them perpendicularly. Knowing the sun's elevation angle at solar noon for your latitude and the target season lets you set the tilt angle of the panels to capture the most energy. A common rule of thumb is to tilt panels at an angle equal to your latitude for year-round optimization, or adjust seasonally — steeper in winter when the sun is lower, shallower in summer. Shading analysis for buildings also requires knowing the sun's elevation to determine when and where shadows will fall on adjacent structures or roof sections.
What day of the year should I use to find the worst-case solar angle for my location?
For locations in the Northern Hemisphere, the winter solstice (around December 21, day 355) produces the lowest solar elevation angle of the year and represents the worst case for solar energy production and the longest shadows. If you are designing a solar installation or conducting a shading study, using the winter solstice elevation guarantees your design is adequate even when the sun is at its lowest. Conversely, day 172 (summer solstice, around June 21) gives the highest elevation and is useful for calculating minimum shadow lengths or maximum solar gain through windows.