Free Sun Angle Calculator

A Solar Angle Calculator (also known as a Sun Position Calculator) is an essential tool for determining the exact position of the Sun in the sky — including the sun elevation angle (or solar altitude), solar azimuth, and other related parameters — for any given location on Earth and any specified time. This free online calculator allows you to input your coordinates, date, and time, and quickly obtain the sun's altitude and azimuth, along with visual charts for a deeper understanding of solar movement.

The Sun's location is most commonly expressed using two angular measurements:

• Elevation (or Altitude) Angle ($\alpha$): The vertical angle between the horizon and the Sun, ranging from $-90^\circ$ (below horizon) to $+90^\circ$ (directly overhead). A value of $0^\circ$ means the Sun is on the horizon.

• Azimuth Angle ($\beta$): The horizontal direction of the Sun measured clockwise from true north, from $0^\circ$ to $360^\circ$. Common bearings include North ($0^\circ$), East ($90^\circ$), South ($180^\circ$), and West ($270^\circ$).

These two angles fully define the Sun's position as seen from a specific point on Earth.

Practical Application: A Concrete Example

To see how a solar angle calculator works in practice, let's take a specific scene. Imagine you are located near the center of the continental United States, at latitude $38.6^\circ$ N and longitude $90.2^\circ$ W (the approximate location of Saint Louis, Missouri). The date is February 20, 2023, the time is 3:00 PM local (Central Standard Time, UTC–6), and you want to know where the Sun is.

Simply enter these parameters into the solar position calculator:

• Latitude: $38.6$ (positive for north)
• Longitude: $-90.2$ (negative for west)
• Date: February 20, 2023
• Time: 15:00
• Time zone: CST (UTC–6)

After processing, the calculator returns a solar elevation angle of $27.37^\circ$ and a solar azimuth angle of $226.98^\circ$. This means the Sun is roughly one-third of the way above the horizon (about 27° high) and is located in the southwest direction (since 227° is south of west). Note: these results can vary slightly from year to year due to calendar and astronomical nuances; the calculator accounts for the selected year.

Optional charts: Many solar angle calculators also provide a graphical display of the Sun's path for the given day — showing elevation and azimuth across time — which can be toggled on or off.

The Mathematical Backbone: Formulas for Solar Elevation and Azimuth

The calculations rely on three principal variables: the observer's latitude ($\phi$), the Sun's declination ($\delta$), and the local hour angle ($\gamma$). The fundamental equations are:

Solar Elevation Angle

$$\sin\alpha = \sin\delta \sin\phi + \cos\delta \cos\phi \cos\gamma$$

Then the elevation is $\alpha = \sin^{-1}(\text{the above})$.

Solar Azimuth Angle

$$\beta = \cos^{-1}\left( \frac{\sin\delta \cos\phi - \cos\delta \sin\phi \cos\gamma}{\cos\alpha} \right)$$

Note that the sign of the hour angle determines whether the azimuth is measured east or west of north; for negative hour angles (morning), the raw $\beta$ is used; for positive hour angles (afternoon), the actual azimuth is $360^\circ - \beta$.

Declination Angle

The declination depends only on the day of the year ($d$), using a simple yet reasonably accurate approximation:

$$\delta = -23.45^\circ \times \cos\left( \frac{360}{365} (d + 10) \right)$$

Here, $d$ starts from 1 on January 1. The value ranges from about $-23.45^\circ$ (December solstice) to $+23.45^\circ$ (June solstice), reflecting the tilt of Earth's axis.

Local Hour Angle

The local hour angle transforms the local solar time ($T$, in hours) into an angle:

$$\gamma = 15^\circ \times (T - 12)$$

This yields values from $-180^\circ$ to $+180^\circ$, crossing $0^\circ$ at solar noon. The $15^\circ/\text{hour}$ factor comes from Earth's rotation rate (360° per 24 hours).

Important Considerations

• Atmospheric Refraction: Light bends as it passes through the atmosphere, making the Sun appear slightly higher than its geometric position. The computed sun elevation angle provided by the calculator is the apparent altitude, which matches what you would actually observe. The difference is most noticeable near the horizon.

• Accuracy of Approximations: The formulas above are simplified; they ignore subtle effects like Earth's elliptical orbit (eccentricity) and precession of the axial tilt. For most practical purposes — solar panels, photography, agriculture, or just curiosity — the accuracy is excellent. A full-featured solar angle calculator uses more precise algorithms (sometimes based on the NREL solar position algorithm) for high-accuracy results.

• Seasonal and Latitudinal Variations: Because declination swings between solstice extremes, the Sun's daily path changes markedly with season. At mid-latitudes, summer brings high noonday elevation and longer days, while winter brings low elevation. Inside the tropics, the Sun passes directly overhead twice a year. In polar regions, the Sun never sets in summer (midnight Sun) and never rises in winter (polar night). The calculator can display the Sun's path for any date, revealing these patterns.