Let’s Make a Sundial!

Let’s make a sundial that tells the hour of the day. I know what you’re thinking. Can’t I just go to my local garden store and buy one? Nope. Those are just for decoration only. A real sundial, one that accurately tells the time, must be built for your location (latitude). All sundials are local.

I recommend reading my previous post on building a solar calendar. Also, if you’ve never given much thought to the apparent motion of the sun across the sky, then check out my post on sidereal time. If you know all about sundials and you just want to build one, then read on!

There are so many different types of sundials. Many European cities have vertical ones built on the walls of older buildings in the town square. They can also be at oblique angles like the equatorial sundial. I have a little ring sundial that you hold up to the sun and peek through a small hole to see the shadow. But here I’m going to describe what most people think of when they picture a sundial, the “London” or horizontal sundial that you see in home gardens.

The oldest known horizontal sundial (pictured below) is about 3,500 years old and made of baked clay.

This is called the “plate” and it would have been placed on a pedestal, on a flat rock, or maybe even on the ground. Notice the hole in the center where the hour angles converge? A stick (called the “gnomon”) would have been inserted there in order to produce a shadow from the sun. It’s really no different than a paper plate sundial that you can create with your kids.

Notice that there is an x-axis line running from left to right through the hole. And a perpendicular line dropping straight down from the hole. That perpendicular line is solar noon. You would set the plate so that the solar noon line is oriented on a north-south line. This line divides morning (the region to the right of the line) from the afternoon (the region to the left of the line). Notice that the sundial is divided into 12 hours. The x-axis marks out 6 AM and 6 PM. The next lines from there (the hour angles) mark out 7 AM and 5 PM, then 8 AM and 4 PM, and so on until you get to solar noon.

Enough History, Let’s Do It

I mentioned earlier that the sundial must be created for your location. You need to know your latitude. In this post I’ll use London which is at 51.5° latitude. Once you have your latitude you can go to any online calculator (here’s a good one) and the angles will be generated for you.

Want to get geeky and do the math yourself? Carl Sabanski has a good primer on the math behind a sundial as does the wiki. The formula for an hour angle H is:

$$tan \space H = sin \space L \space tan(15 * t)$$

or:

$$H = arctan (sin \space L * tan (15 * t))$$

This is implemented in C# as follows:

double lat = 51.5; 
double L = Math.Sin(lat * 2 * Math.PI / 360);

for (int t = -6; t <= 6; t++)
{
    double h = t * 15.0;
    double H = Math.Atan(L * Math.Tan(h * 2 * Math.PI / 360)) * 360 / (2 * Math.PI);

    if (t == 0)
    {
        Console.WriteLine(t + ": " + H + " (Solar Noon)");
    }
    else
    {
        Console.WriteLine(t + ": " + H);
    }
}

If I put this code in a console app it generates the following angles:

• 90 (6 AM / 6 PM)
• 71 (7 AM / 5 PM)
• 54 (8 AM / 4 PM)
• 38 (9 AM / 3 PM)
• 24 (10 AM / 2 PM)
• 12 (11 AM / 1 PM)
• 0 (Solar Noon)

Using a protractor, I can draw this out on paper in the same way an ancient Egyptian mathematician drew the one on the clay tablet above:

I left off the afternoon hour lines but you get the idea. From here you can DIY a sundial made out of wood or clay. This site shows you step-by-step how to make one out of scrap wood. Enjoy!