Sundial Gnomon Angle Calculator

The Ancient Art of Telling Time by Shadows

For millennia before mechanical clocks, humanity told time by observing the movement of shadows cast by the sun. Sundials remain among the most elegant astronomical instruments ever devised, requiring no power source and continuing to function for centuries with proper construction. At the heart of every sundial lies the gnomon—the shadow-casting element whose angle must be precisely calculated based on the observer's latitude. This calculator determines the correct gnomon angle, hour line positions, and other geometric parameters needed to design an accurate sundial for any location on Earth.

Understanding the Celestial Sphere

To design a sundial, one must first understand how the sun appears to move across the sky. From our perspective on Earth, the sun traces an arc from east to west each day, with the arc's height and position varying with the seasons. At the equinoxes, the sun rises due east and sets due west, tracing a path directly along the celestial equator. During summer in the Northern Hemisphere, the sun's path is higher and longer; during winter, it is lower and shorter. The gnomon of a properly designed sundial must be aligned parallel to Earth's axis of rotation, pointing toward the celestial pole, so that the sun's apparent motion translates directly into the movement of the shadow across the hour lines.

The Gnomon Angle and Latitude

The most critical measurement for any sundial is the gnomon angle, which equals the observer's geographic latitude. At the North Pole (90° latitude), the gnomon would point straight up; at the equator (0° latitude), it would lie flat along the ground pointing north. For most locations, the gnomon angle falls somewhere between these extremes. A sundial in London (approximately 51.5° N) requires a gnomon tilted 51.5° from horizontal, while one in Miami (approximately 25.8° N) needs a shallower 25.8° angle. This relationship holds because a gnomon parallel to Earth's axis effectively points at the celestial pole around which the sun appears to rotate.

Types of Sundials

While the gnomon angle remains constant for a given latitude, sundials come in many configurations that affect how hour lines are calculated. The horizontal sundial, the most common type, features a flat dial plate with the gnomon rising at the latitude angle. Vertical sundials mount on walls and require different hour line calculations depending on which direction the wall faces. Equatorial sundials have their dial plate tilted parallel to the equator, producing evenly spaced hour lines but requiring seasonal plate switching. Armillary sundials use multiple rings to represent the celestial sphere, while analemmatic sundials replace the fixed gnomon with a movable vertical marker that must be repositioned for each date.

Calculating Hour Lines for Horizontal Sundials

For a horizontal sundial, the angle of each hour line from the noon line is given by the formula: tan(H) = tan(15° × t) × sin(φ), where H is the hour line angle, t is the number of hours from noon, and φ is the latitude. Because this formula involves the sine of latitude, the hour lines are not evenly spaced—they bunch together near noon and spread apart toward morning and evening. At higher latitudes, this effect is more pronounced. The calculator computes these angles for each hour, allowing you to lay out the dial face accurately.

The Equation of Time

A perfectly constructed sundial tells apparent solar time, not the mean solar time shown on clocks. The difference between these two times is called the equation of time, which varies throughout the year by as much as 16 minutes. This variation arises from two factors: Earth's elliptical orbit causes it to move faster when closer to the sun and slower when farther away, and Earth's axial tilt means the sun's apparent motion has a component perpendicular to the celestial equator. Some elaborate sundials incorporate an analemma—a figure-eight curve—to correct for this difference, but most users simply accept the discrepancy or consult a correction table.

Longitude Correction

Standard time zones are based on specific meridians, typically 15° apart, but most locations do not sit precisely on these meridians. A sundial shows local solar time, which differs from zone time by four minutes for every degree of longitude east or west of the zone's reference meridian. For example, if you live 5° west of your time zone's central meridian, your sundial will read 20 minutes ahead of your clock at solar noon. This fixed offset can be incorporated into the dial design by rotating the hour lines slightly, or the user can mentally adjust the reading.

Designing the Dial Plate

Once you know the gnomon angle and hour line positions, you can design the dial plate itself. The plate should be made of durable material—stone, metal, or weather-resistant wood—and inscribed with hour lines radiating from the gnomon's base. The noon line always points true north (in the Northern Hemisphere) or true south (in the Southern Hemisphere). Hour lines are typically numbered from 6 AM to 6 PM, though some dials include lines for dawn and dusk hours that are only illuminated near the solstices. Decorative elements such as zodiac symbols, seasonal arcs, or inspirational mottoes are traditional additions.

Gnomon Construction

The gnomon itself must be precisely angled and sturdy enough to resist wind and weather. It can be a simple triangular plate with the hypotenuse at the latitude angle, or a rod or edge that casts a thin shadow for precise reading. The style—the shadow-casting edge—should be straight and sharp for accuracy. Some gnomons include a nodus, a small sphere or point along the style that casts a spot of light or shadow used to indicate the date as well as the time. The height of the gnomon relative to the dial plate determines the length of the shadow and thus the usable size of the dial.

Seasonal Variations

Throughout the year, the sun's declination changes from about +23.5° at the summer solstice to -23.5° at the winter solstice. This affects the length and position of shadows. Some sundials include date lines—hyperbolic curves crossing the hour lines—that show the time of year. The equinox lines are straight, while the solstice lines curve in opposite directions. By reading where the tip of the shadow falls relative to these curves, one can determine not only the time but also the approximate date.

Accuracy Considerations

A well-made sundial can be accurate to within a minute if properly aligned and if the equation of time and longitude corrections are applied. Alignment is critical: the dial plate must be perfectly level (for horizontal sundials) or plumb (for vertical sundials), and the gnomon must point exactly toward true north, not magnetic north. The difference between true north and magnetic north—called magnetic declination—varies by location and changes over time. Use a compass corrected for local declination, or sight along Polaris at night, to establish true north before installing your sundial.

Historical Significance

Sundials were the primary timekeeping devices of ancient civilizations. The Egyptians used obelisks as gnomons as early as 3500 BCE. The Greeks refined sundial design, introducing the hemispherical dial and understanding the relationship between gnomon angle and latitude. Medieval Islamic astronomers created sophisticated portable sundials and universal instruments. In Europe, sundials adorned churches, public squares, and private gardens, often bearing Latin inscriptions reminding viewers of the passage of time. Even after mechanical clocks became widespread, sundials remained popular as both functional instruments and symbols of cosmic order.

Modern Sundial Making

Today, sundial making combines ancient craft with modern precision. Computer-aided design allows for exact calculation of hour lines, and laser cutting produces gnomons with perfectly sharp edges. Materials range from traditional bronze and stone to modern stainless steel and acrylic. Some contemporary sundials incorporate digital elements, using the shadow position to trigger electronic displays, while others embrace minimalist aesthetics with clean lines and geometric forms. The calculator provided here enables anyone to design a mathematically correct sundial, bringing this ancient technology into the present.

Using the Calculator

Enter your latitude in degrees (positive for north, negative for south) and select the sundial type. The calculator computes the gnomon angle, which equals your latitude for horizontal and equatorial dials, and provides the hour line angles measured from the noon line. For a horizontal sundial, these angles tell you where to draw each hour line on the dial plate. The calculator also provides the longitude correction factor if you enter your longitude and time zone, allowing you to adjust the dial for standard time. With these values in hand, you can proceed to construct a sundial that will tell accurate solar time for generations to come.

Practical Construction Tips

When building your sundial, start by creating a full-size template on paper or cardboard. Use a protractor to lay out the hour lines precisely, measuring each angle from the noon line. Cut the gnomon from rigid material, ensuring the shadow-casting edge is straight. A right triangle with one acute angle equal to your latitude makes an ideal gnomon shape. Install the dial on a stable, level surface with the noon line pointing true north. Check the alignment at solar noon—when the shadow falls exactly on the noon line—and adjust if necessary. With care and patience, you can create a beautiful and functional timepiece that connects you to centuries of astronomical tradition.