How a Sundial Works and Why Its Angle Changes With Your Latitude

A sundial does not measure the Sun's position in the sky. It measures the shadow cast by a rod that points at a fixed point in space, the celestial north pole, the same point that Polaris sits near. That rod is called the gnomon, and the angle it makes with the horizontal ground is the whole secret of why a sundial works at all.

Most decorative sundials sold in garden centres are set flat, with a vertical gnomon at ninety degrees to the dial face. Those work nowhere accurately. They are furniture. A working sundial tilts its gnomon so that it runs parallel to Earth's rotational axis. Because Earth spins on that axis at a steady rate, one full rotation every twenty-four hours, the shadow sweeps the dial face at a constant speed, one hour per fifteen degrees of arc.


Earth is a sphere, and your position on that sphere changes the angle at which the sky's pole sits above your horizon. At the equator, the celestial north pole lies exactly on the horizon, zero degrees up. At the North Pole, it is directly overhead, ninety degrees up. Everywhere between, the pole sits at an elevation equal to your latitude.


Bengaluru sits at roughly 13 degrees north. A sundial there needs a gnomon angled at 13 degrees above the horizontal. Mumbai is at about 19 degrees north, gnomon at 19 degrees. Delhi, near 29 degrees north, gnomon at 29 degrees. Move a sundial built for Delhi to Kochi, which sits near 10 degrees north, and every shadow falls wrong by about nineteen degrees of error. The dial face itself also needs to be re-engraved, because the hour lines fan out from the gnomon's base at angles that are calculated from the latitude.


The formula for each hour line is a standard trigonometric relationship: the tangent of the hour-line angle equals the sine of the latitude multiplied by the tangent of the hour angle from solar noon. Change the latitude input and every line on the face shifts.

Even a perfectly built, correctly angled sundial will disagree with your phone's clock on most days of the year. This disagreement has a name, the equation of time, and it runs as large as sixteen minutes fast or fourteen minutes slow depending on the calendar date.

Two things cause it. First, Earth's orbit around the Sun is an ellipse, not a perfect circle. Earth moves faster when it is closer to the Sun in January and slower when it is farther away in July. A clock divides the year into equal seconds; the Sun does not cooperate. Second, Earth's rotational axis is tilted at 23.5 degrees relative to its orbital plane. This tilt, which gives us seasons, also means the Sun's apparent east-west speed across the sky varies through the year.

The equation of time was known to Indian astronomers centuries before European clockmakers formalised it. The Surya Siddhanta, a Sanskrit astronomical text, carried calculations for the Sun's uneven motion, what it called manda correction, that account for the same orbital eccentricity a modern ephemeris uses.

The largest sundial in the world still in use is the Samrat Yantra at Jantar Mantar in Jaipur, built by Maharaja Jai Singh II in the early eighteenth century. Its gnomon is a right-triangle staircase 27 metres tall, its hypotenuse angled at precisely 27 degrees, Jaipur's latitude. The curved quadrant scales on either side are graduated finely enough to read time to two seconds of accuracy.

Jai Singh built five Jantar Mantar observatories across northern India, Jaipur, Delhi, Varanasi, Ujjain, Mathura, and each Samrat Yantra carries a gnomon angled to match the latitude of its city. Delhi's gnomon is shallower than Jaipur's because Delhi sits a fraction further north. This is not a detail of historical interest. It is the same geometric rule that governs a garden sundial: the instrument only works where it was built for.

When the gnomon points at the celestial pole, it becomes parallel to Earth's spin axis. The Sun, from the perspective of someone standing on Earth's surface, appears to rotate around that axis once a day. The shadow the gnomon casts onto the dial face therefore sweeps at the same angular rate as Earth's rotation, fifteen degrees per hour, every hour, all year.

This constancy is what makes a sundial a clock rather than just a shadow. The shadow's direction encodes the hour; the shadow's length encodes the season, growing longer in winter when the Sun is low and shorter in summer when it is high. A horizontal dial gives you both pieces of information simultaneously, which is why the instrument survived for millennia before mechanical clocks arrived to replace it.

The gnomon's angle is not a quirk of design. It is the one geometric fact that connects a fixed rod in a garden to the rotation of a planet, and it only holds for the latitude where it was set.