Sun Angle Calculator

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Reviewed by: David Chen, P.E. (Professional Engineer)

Determine the precise position of the sun—its altitude (elevation) and azimuth (direction)—at any given location, date, and time. Essential for solar power design, passive heating, and architectural planning.

Sun Angle Calculator

Solar Altitude Angle ($\alpha$): —

Solar Azimuth Angle ($\gamma$): —

Detailed Calculation Steps

Enter the values and click ‘Calculate’ to see the step-by-step process.

Sun Angle Calculator Formula

Solar Declination ($\delta$):

$$\delta = 23.45^\circ \cdot \sin\left(\frac{360}{365} \cdot (N – 81)\right)$$

Solar Zenith Angle ($\theta_z$):

$$\cos(\theta_z) = \sin(\phi) \cdot \sin(\delta) + \cos(\phi) \cdot \cos(\delta) \cdot \cos(\omega)$$

Solar Altitude Angle ($\alpha$):

$$\alpha = 90^\circ – \theta_z$$

Formula Sources: NREL Solar Resource Models, Energy.gov Solar Basics

Variables

The calculator requires the following variables for an accurate solar angle determination:

  • Latitude ($\phi$): The geographic latitude of the location in degrees (North is positive, South is negative).
  • Longitude ($\lambda$): The geographic longitude of the location in degrees (East is positive, West is negative).
  • Time Zone Offset (UTC $\pm$ hours): The difference in hours between the local time zone and Coordinated Universal Time (UTC).
  • Date (Local): The specific calendar day for the calculation, used to determine the Day of the Year ($N$) and thus the Declination Angle ($\delta$).
  • Time (Local HH:MM): The specific local time of day, used with the location data to find the Local Solar Time (LST).

What is Sun Angle Calculator?

The sun angle calculator is a tool used in fields like solar energy, architecture, and astronomy to determine the precise position of the sun in the sky relative to a specific location on Earth at a given moment. This position is typically defined by two primary angles: the Solar Altitude Angle ($\alpha$) and the Solar Azimuth Angle ($\gamma$).

The Solar Altitude Angle (or Solar Elevation Angle) is the vertical angle between the sun’s center and the local horizon, ranging from $0^\circ$ (at the horizon) to $90^\circ$ (directly overhead). The Solar Azimuth Angle is the horizontal angle of the sun’s position, measured clockwise from due North.

Knowing these angles is crucial for optimizing the placement of solar panels, designing buildings for passive heating/cooling (by managing sun exposure), and planning outdoor activities that depend on shade or sunlight throughout the day.

How to Calculate Sun Angle (Example)

Let’s use an example to walk through the general process (simplified):

  1. Determine Location and Date Variables: Input Latitude ($\phi=34.05^\circ$), Longitude ($\lambda=-118.24^\circ$), Date (June 21), and Time (12:00 PM).
  2. Calculate Day of the Year ($N$): June 21 is the 172nd day of the year.
  3. Find Declination Angle ($\delta$): Using the formula with $N=172$, we find $\delta \approx +23.45^\circ$ (Summer Solstice).
  4. Calculate Local Solar Time (LST) and Hour Angle ($\omega$): Adjust the local time for the Equation of Time (EOT) and Longitude correction to find LST. For LST = 12:00 (solar noon), the Hour Angle $\omega = 0^\circ$.
  5. Calculate Zenith Angle ($\theta_z$): Plug $\phi=34.05^\circ$, $\delta=23.45^\circ$, and $\omega=0^\circ$ into the Zenith formula. $\cos(\theta_z) = \sin(34.05)\sin(23.45) + \cos(34.05)\cos(23.45)\cos(0)$, yielding $\theta_z \approx 10.60^\circ$.
  6. Determine Altitude Angle ($\alpha$): $\alpha = 90^\circ – 10.60^\circ = 79.40^\circ$. (The sun is high in the sky).

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Frequently Asked Questions (FAQ)

Is the time zone offset important for the calculation?

Yes, absolutely. The time zone offset is crucial because the solar angle depends on the sun’s position relative to your meridian. The offset, along with your longitude, is used to accurately convert your local clock time to the true Local Solar Time (LST), which dictates the sun’s position.

What is the difference between Solar Altitude and Solar Azimuth?

Solar Altitude is the vertical angle (how high the sun is above the horizon), while Solar Azimuth is the horizontal angle (the compass direction of the sun), typically measured from true North.

Why does the calculation include the Day of the Year ($N$)?

The Day of the Year ($N$) determines the Solar Declination Angle ($\delta$), which is the angle of the sun’s rays relative to the Earth’s equator. This angle changes daily due to the Earth’s axial tilt and orbit, and it is what causes seasons.

Can this calculator determine sunrise and sunset times?

While the calculator provides the sun’s position at a specific time, it does not directly calculate sunrise/sunset. Those times occur when the Solar Altitude Angle ($\alpha$) is $0^\circ$, requiring an iterative or specialized solution, which is beyond the scope of this primary calculation.

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