Sun Angle Calculator
Calculate the sun’s position (altitude and azimuth) for any location and time.
Enter Your Parameters
sin(Altitude) = sin(Declination)sin(Latitude) + cos(Declination)cos(Latitude)cos(Hour Angle)
Chart showing the sun’s Altitude and Azimuth angles throughout the selected day.
| Time of Day | Solar Altitude (°) | Solar Azimuth (°) |
|---|
Table of sun positions at key times for the selected location and date.
All About the Sun Angle Calculator
What is a sun angle calculator?
A sun angle calculator is a powerful tool used to determine the precise position of the sun in the sky at any given moment, for any location on Earth. It calculates two primary coordinates: the solar altitude and the solar azimuth. The altitude is the sun’s vertical angle above the horizon (0° at sunrise/sunset, up to 90° directly overhead), while the azimuth is its compass direction (e.g., 0°/360° for North, 90° for East, 180° for South, 270° for West). This sun angle calculator is indispensable for professionals in various fields, including solar energy system design, architecture, photography, filmmaking, and agriculture.
Anyone who needs to understand the sun’s path and its impact on light and shadow will find a sun angle calculator useful. A common misconception is that the sun is always highest in the sky at 12:00 PM noon. In reality, “solar noon” (when the sun reaches its peak altitude) rarely aligns perfectly with clock noon due to time zones and the Earth’s elliptical orbit, a factor this calculator accurately accounts for.
Sun Angle Formula and Mathematical Explanation
The calculations performed by this sun angle calculator are based on established principles of spherical astronomy. Here is a step-by-step breakdown of the core formulas:
- Solar Declination (δ): This is the angle between the sun’s rays and the plane of the Earth’s equator. It varies throughout the year from approximately -23.45° on the winter solstice to +23.45° on the summer solstice. A common approximation is:
δ = -23.45 * cos( (360/365) * (N + 10) )
where ‘N’ is the day of the year. - Equation of Time (EoT): This corrects for the discrepancy between apparent solar time and mean solar time (clock time). It’s a complex value, but can be approximated.
- Time Correction (TC): This factor adjusts local clock time to solar time, accounting for both the Equation of Time and the difference in longitude from the time zone’s standard meridian.
TC = 4 * (Standard Longitude - Local Longitude) + EoT - Hour Angle (HRA or ω): This measures the sun’s angular distance from the local solar noon, converted to degrees. It is 0° at solar noon, negative in the morning (East), and positive in the afternoon (West).
HRA = 15 * (Local Solar Time - 12) - Solar Altitude (α): This is the primary output, calculated using the latitude (φ), declination (δ), and hour angle (ω).
sin(α) = sin(δ)sin(φ) + cos(δ)cos(φ)cos(ω) - Solar Azimuth (γ): This is the compass direction, also derived from the same variables. The formula is more complex and requires careful handling of quadrants to get the correct 0-360° value.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| φ (phi) | Geographic Latitude | Degrees | -90 to +90 |
| λ (lambda) | Geographic Longitude | Degrees | -180 to +180 |
| N | Day of the Year | Integer | 1 to 366 |
| δ (delta) | Solar Declination | Degrees | -23.45 to +23.45 |
| ω (omega) | Hour Angle | Degrees | -180 to +180 |
| α (alpha) | Solar Altitude Angle | Degrees | 0 to 90 (during daytime) |
| γ (gamma) | Solar Azimuth Angle | Degrees | 0 to 360 |
Practical Examples (Real-World Use Cases)
Example 1: Solar Panel Installation in Phoenix, Arizona
An engineer is designing a rooftop solar array in Phoenix (Latitude: 33.45°, Longitude: -112.07°). They want to find the sun’s position on the summer solstice (June 21st, Day 172) at solar noon to determine the maximum potential energy generation.
- Inputs: Latitude=33.45, Longitude=-112.07, Day=172, Time=12.7 (approx. solar noon), Timezone=-7.
- Outputs from the sun angle calculator:
- Solar Altitude: ~79.9°
- Solar Azimuth: ~180° (due South)
- Interpretation: At its peak on the longest day of the year, the sun will be very high in the sky (almost 80°). This confirms that tilting panels at a low angle (e.g., 10-20°) would be optimal for summer performance. The azimuth of 180° confirms the panels should face true South for maximum exposure. For more detailed planning, you might consult a solar panel efficiency calculator.
Example 2: Photography in London, UK
A photographer wants to capture a photo of Tower Bridge during the “golden hour” in winter. They are planning a shoot on December 15th (Day 349) near sunset in London (Latitude: 51.5°, Longitude: -0.1°, Timezone: 0).
- Inputs: Latitude=51.5, Longitude=-0.1, Day=349, Time=15.5 (3:30 PM), Timezone=0.
- Outputs from the sun angle calculator:
- Solar Altitude: ~2.5°
- Solar Azimuth: ~225° (South-West)
- Interpretation: The sun angle calculator shows that at 3:30 PM, the sun will be very low on the horizon (2.5°), creating the long shadows and warm light characteristic of golden hour. The azimuth of 225° tells the photographer exactly where to stand relative to the bridge to get the sun in their shot. This precise information is far more reliable than just guessing.
How to Use This Sun Angle Calculator
Using our sun angle calculator is straightforward. Follow these steps for an accurate reading:
- Enter Latitude: Input the latitude of your location in decimal degrees. Positive values are for the Northern Hemisphere, and negative values are for the Southern Hemisphere.
- Enter Longitude: Input the longitude. Positive values are for East of the Prime Meridian, and negative values are for West.
- Enter Day of the Year: Provide the day number (1 for January 1st, 365 for December 31st).
- Enter Local Time: Use a 24-hour format. For minutes, use decimals (e.g., 9:45 AM is 9.75).
- Enter Timezone: Input the location’s offset from Coordinated Universal Time (UTC).
As you enter the data, the results will update in real-time. The “Solar Altitude Angle” is the main result, showing how high the sun is. The “Solar Azimuth Angle” tells you its compass direction. The chart and table provide a broader view of the sun’s path throughout the day, which is crucial for comprehensive planning. Understanding these outputs helps in making informed decisions, whether you’re optimizing a building’s design or planning a project timeline.
Key Factors That Affect Sun Angle Results
Several key factors influence the sun’s position. Our sun angle calculator accounts for all of these:
- Latitude: This is the most critical factor. Your north-south position on Earth determines the fundamental path of the sun across the sky throughout the year. Locations closer to the equator experience higher sun angles year-round.
- Day of the Year: The Earth’s axial tilt (~23.45°) is the reason for the seasons. This tilt causes the sun’s declination to change daily, drastically altering sun angles between summer and winter.
- Time of Day: This determines the sun’s position along its daily arc from sunrise in the east to sunset in the west. The hour angle is directly derived from the time of day.
- Longitude and Timezone: These two factors work together to align your local clock time with the actual solar time. The sun is highest at “solar noon,” which can be significantly different from 12:00 PM on your watch, a discrepancy our sun angle calculator corrects.
- Atmospheric Refraction: While not explicitly a user input, it’s a physical phenomenon. The atmosphere bends sunlight, making the sun appear slightly higher than it geometrically is, especially near the horizon. Advanced models account for this.
- Elevation/Altitude: Being at a higher elevation can slightly change the time of sunrise and sunset, as you can “see” over the horizon. For most standard uses, this effect is minor but can be relevant for astronomical observations. For financial planning related to projects, consider using a budget calculator.
Frequently Asked Questions (FAQ)
1. Why is the sun not at its highest point at 12:00 PM?
This is due to two main reasons: the designation of time zones, which create a standardized time across a wide band of longitudes, and the Equation of Time, which accounts for the Earth’s non-circular orbit. Our sun angle calculator correctly finds “solar noon” for you.
2. What is the difference between altitude and elevation?
In this context, “altitude” (or solar altitude angle) refers to the sun’s angle above the horizon. “Elevation” refers to your height above sea level. They are different concepts.
3. How is azimuth measured?
Azimuth is typically measured in degrees clockwise from true North. So, North is 0° or 360°, East is 90°, South is 180°, and West is 270°. This is the standard used by our sun angle calculator.
4. Can this calculator predict sunrise and sunset times?
While this tool focuses on calculating the sun’s position at a specific time, you can find sunrise or sunset by finding the time when the solar altitude is 0°. For dedicated tools, you might search for a sunrise/sunset calculator.
5. Does this calculator account for Daylight Saving Time (DST)?
You must account for DST manually in the “Timezone” input. For example, if a location is normally UTC-5 but is observing DST, it becomes UTC-4. You should enter -4 as the timezone.
6. How accurate is this sun angle calculator?
This sun angle calculator uses standard, widely accepted astronomical formulas. For most practical purposes (architecture, solar, photography), the accuracy is very high, typically within a fraction of a degree.
7. What does a negative altitude angle mean?
A negative altitude angle means the sun is below the horizon. This occurs before sunrise and after sunset. The calculator will show these values if you input a time during the night.
8. Can I use this for locations in the Southern Hemisphere?
Yes. Simply enter a negative value for the latitude. The sun angle calculator will correctly compute the sun’s path, where the sun transits through the northern part of the sky instead of the southern.
Related Tools and Internal Resources
Explore other tools and resources that can help with your planning and calculations:
- Date Duration Calculator: Calculate the number of days between two dates, useful for finding the “Day of the Year” input for this calculator.
- Time Zone Converter: A helpful tool for ensuring you are using the correct UTC offset, especially when dealing with Daylight Saving Time.
- Solar Panel Angle Calculator: A specialized tool that uses sun angle data to recommend the optimal tilt for solar panels based on your location and goals.