Printable Sundials, Explained
This guide shows you why sundial hour lines are unevenly spaced, how to compute each line from your latitude, and how to read the dial once the sun starts casting shadows on it.
What a sundial actually measures
A sundial tells time by the direction of a shadow. As the Earth turns, the sun appears to sweep across the sky at a steady 15 degrees per hour (360^\circ / 24 \text{ h}). A rod tilted parallel to the Earth's axis, called the gnomon or style, casts a shadow that rotates with the sun. The trick is that the shadow does not rotate evenly across a flat dial face. It bunches up near noon and spreads out toward morning and evening, and the exact spacing depends on how far north or south you stand.
Here is the hook. Build a dial for latitude 40° and the 3 p.m. line sits about 32.7° from the noon line. Take that same paper to latitude 50° and the true 3 p.m. shadow lands at 39.6°, nearly 7 degrees off. The dial is silently wrong. A sundial is a piece of geometry tuned to one place, and this generator retunes the geometry to your latitude.
When to build one, and when not
A printable dial is ideal for a classroom, a garden marker, or a demonstration of how solar geometry works. Print it on cardstock, fold the gnomon, and you have a working instrument in ten minutes. It is accurate to a few minutes if you level it and align it well.
Do not expect it to match your phone clock exactly. A sundial reads apparent solar time. Three separate effects push that away from civil clock time: the equation of time (up to about ±16 minutes over the year), your longitude offset inside your time zone (4 minutes per degree), and daylight saving (a full hour). If you need clock time to the minute, use a clock. If you want to understand where clock time comes from, build the dial.
The gnomon does not point straight up. It must tilt to match your latitude and aim at the celestial pole, which is true north in the northern hemisphere. That tilt is what makes the shadow track the sun's daily circle correctly.
The hour-line formula and the intuition
For a horizontal (tabletop) dial, the angle of each hour line from the noon line is given by:
Here \theta is the angle of the hour line measured from the noon line on the dial face, \phi is your latitude, and h is the number of hours before or after solar noon (so 1 p.m. is h = 1, 3 p.m. is h = 3, 9 a.m. is h = -3). The term 15^\circ \times h is the sun's hour angle: how far it has swung from due south.
The intuition is in the \sin(\phi) factor. If you stood at the North Pole (\phi = 90^\circ), \sin(\phi) = 1, and the hour lines would be perfectly even at 15° apart, because the dial face lies flat under a sun that circles the horizon uniformly. At the equator (\phi = 0^\circ), \sin(\phi) = 0 and a horizontal dial collapses (all lines fold onto the noon line); there you would use a vertical or equatorial dial instead. Every latitude between compresses the morning and evening lines by the factor \sin(\phi).
A vertical south-facing dial uses the same equation with \cos(\phi) in place of \sin(\phi):
That single swap is why a wall dial and a tabletop dial for the same city look different. At 40°, \sin(40^\circ) = 0.643 but \cos(40^\circ) = 0.766, so the vertical dial spreads its hour lines wider.
A worked example at latitude 40 degrees
Reproducing the demo: horizontal dial, 40° north
The demo defaults build a horizontal dial for latitude 40°. Work the 3 p.m. line by hand.
- The hour angle is 15^\circ \times 3 = 45^\circ, so \tan(45^\circ) = 1.
- Multiply by \sin(40^\circ) = 0.6428: the product is
0.6428. - Take the arctangent: \theta = \arctan(0.6428) = 32.74^\circ from the noon line.
Now the 1 p.m. line. The hour angle is 15^\circ, \tan(15^\circ) = 0.2679, times 0.6428 gives 0.1722, and \arctan(0.1722) = 9.77^\circ. Notice the gap from noon to 1 p.m. is only 9.77°, but the gap from 2 p.m. to 3 p.m. is 32.74^\circ - 19.64^\circ = 13.1^\circ. The lines fan out as you move from noon. The gnomon edge for this dial tilts up at exactly 40°, equal to the latitude.
| Solar hour | h | Hour angle | Angle θ (horizontal) | Angle θ (vertical) |
|---|---|---|---|---|
| 12 noon | 0 | 0° | 0.00° | 0.00° |
| 1 p.m. | 1 | 15° | 9.77° | 11.66° |
| 2 p.m. | 2 | 30° | 19.64° | 23.13° |
| 3 p.m. | 3 | 45° | 32.74° | 37.43° |
| 4 p.m. | 4 | 60° | 48.07° | 53.11° |
| 5 p.m. | 5 | 75° | 67.38° | 70.44° |
The morning hours mirror these: 11 a.m. is -9.77°, 9 a.m. is -32.74°, and so on.
How latitude reshapes the fan of lines
The chart below plots the horizontal hour-line angle for the afternoon hours at three latitudes. Watch how the whole fan opens up as latitude increases: at 20° the lines huddle near noon, at 60° they spread almost to the even 15°-per-hour ideal.
Reading and aligning the finished dial
Once printed, three physical steps decide accuracy. First, level the dial face. Second, tilt the gnomon to your latitude for a horizontal dial (or its complement, 90^\circ - \phi, for a vertical one). Third, aim the noon line at true north, not magnetic north. Magnetic declination can be several degrees off; a 5-degree alignment error shifts every reading by about 20 minutes.
Now convert what you read to clock time. Start from the shadow's solar time, then apply three corrections:
- Equation of time
- A yearly wobble from Earth's elliptical orbit and axial tilt. It ranges from about
-14minutes in mid-February to+16minutes in early November, and is zero around mid-April, mid-June, September 1 and December 25. - Longitude offset
- Your time zone is centered on one meridian. The sun crosses
1°of longitude every 4 minutes. If you are5°west of your zone's central meridian, solar noon arrives 20 minutes late. - Daylight saving
- A flat
+1hour during summer months. The dial knows nothing about legislation.
The generator's optional longitude correction folds the longitude offset directly into the hour-line placement, so the dial reads closer to clock time. The equation of time cannot be built into fixed lines because it changes daily; print a small correction table beside the dial instead.
Common mistakes
The single biggest error is printing at the wrong scale. If your printer shrinks the page to "fit," the hour angles stay correct but the gnomon fold template no longer matches the dial center. Print at 100% (no scaling) and check a known dimension with a ruler before folding.
A second mistake is reusing a dial across latitudes. The angles in the table above are correct only at 40°. Moving 10 degrees north introduces errors of several degrees on the outer lines. Rebuild for each location.
A third is confusing the gnomon's tilt with its shape. Only the upper edge of the gnomon, the straight edge parallel to the Earth's axis, casts the time-telling shadow. That edge must sit over the dial center and rise at the latitude angle. A gnomon that is the right height but the wrong angle reads wrong all day.
Finally, do not expect the dial to work outside its hemisphere or at hours when the sun is behind the face. A south-facing vertical dial cannot read early morning or late evening in summer, when the sun swings around to the north.
Related tools
If you are printing templates that must come out at true physical size, the same 100%-scale discipline applies to the Papercraft Box Generator, which lays out box nets with fold lines and glue tabs. For laying out your own dial measurements or checking print scale, the Graph Paper Generator prints exact-scale grids, including a polar grid that suits radial hour lines. If your project turns an image into a stitched chart, see the Cross-Stitch Pattern Maker, and for tension and sizing math in yarn work, the Knitting & Crochet Gauge Converter.
Frequently asked questions
Why are my sundial's hour lines not evenly spaced?
They should not be. Only an equatorial dial has even 15° spacing. On a horizontal or vertical dial the spacing follows \tan(\theta) = k \cdot \tan(15^\circ h), which bunches the lines near noon. At 40° the noon-to-1 gap is 9.77° while the 4-to-5 gap is 19.3°.
Why does my sundial disagree with my phone by a quarter hour?
Three reasons stack up: the equation of time (up to about 16 minutes), your longitude within the time zone (4 minutes per degree), and daylight saving (a full hour). A 15-minute gap is entirely normal and correct for the sundial.
Can I use a dial made for 40 degrees at 34 degrees?
Not accurately. At 34° the 3 p.m. line should be at 27.9°, but the 40° dial puts it at 32.7°, an error near 5 degrees. Enter your own latitude and rebuild.
How do I find true north without a compass?
At solar noon (corrected for longitude and the equation of time) the sun is due south in the northern hemisphere, so shadows point due north. Mark that shadow and align the dial's noon line to it. This avoids magnetic declination entirely.
What latitude should I type if I only know my city?
Use your city's latitude to one decimal place. A 0.1° error in latitude shifts the outer hour lines by only a fraction of a degree, well below alignment error, so approximate values are fine.
Related tools
Cross-Stitch Pattern Maker
Turn any picture into a cross-stitch or perler-bead chart with symbols and a DMC thread legend.
String Art Generator
Turn a photo into string art: one continuous thread wound around nails, with the exact nail sequence and a printable template.
Graph Paper Generator
Print exact-scale graph paper: square, dot, isometric, hex, lined, polar and log grids.
Knitting & Crochet Gauge Converter
Convert stitch and row counts between gauges, or size a project from your own tension.
Papercraft Box Generator
Generate printable box templates (nets) with fold lines and glue tabs at true physical size.