Draw a vertical line, now draw a line perpendicular (AB) to the vertical line. (Fig. #1)
Now draw a line from A through the vertical line so that angle B'AB is equal to the latitude of the place where the dial is to be used. Remember this angle for making the gnomon. (Fig. #2)
Draw a line perpendicular to line AB' that intersects line AB at point B. (Fig. #3)
Now use the distance A'B to find C (use the compass), draw a semi-circle centered at C and divide it into 15 degree pieces (use the protractor). (Fig. #4)
Why 15 degrees? One day is 24 hours, or the amount of time it take for "the sun to go around the earth". So in 24 hours the sun "moves" through 360 degrees (recall that there are 360 degrees in a circle). Therefore to find how many degrees are in 1 hour we divide 360 by 24. 360/24=15 degrees, this is called the hour angle of the sun.
Connect the 15 degree points along line BB' to point A. These are the hour lines for your sundial. Draw a line through A and parallel to line BB', this is the 6:00 hour line. (Fig. #5)
To find the morning hour line "flip" the afternoon hour lines over the 6 hour line. In other words angle B-A-11 is equal to angle B-A-1, likewise angle B-A-10 is equal to angle B-A-2 etc. (Fig. #6)
All that is left is to add the gnomon and set the dial in the sun. The gnomon is a triangular piece that has one angle equal to the latitude of the place where the dial is to be used. See the diagram below on how to afix the gnomon to the dial.
It's now time to take the dial outside and start to tell time. Orient your dial so that the 12:00 hour line and gnomon (which lies along the 12:00 line) are pointing toward TRUE NORTH.
Point of interest: The above procedure for graphically making a horizontal sundial was used by James Madison. He designed a dial using this same formula while an undergraduate at Princeton circa 1769. The image below is the actual work done by James Madison.
NOTE: The faint image of the Sun at the center of the document and the faint concentric circles are Madison's notes about the Copernican System which are on the other side of the sundial calculation page.
If you live in the Northern hemisphere finding true north is as easy as finding the North star and setting your dial to point at it, also if you measure the angle of the North star (Polaris) above the horizon you will have your latitude. DO NOT use a compass to find north, a compass shows magnetic north NOT true north.
Another way to find true north is to watch the shadow of a vertical rod, when the shadow is at its shortest length it is pointing true North/South.