13-Pointed Star: Approximate Construction

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, Universityof Wisconsin - Green Bay

A 13-pointed star pushes the limits of practical construction pretty hard. You will end up with a wedge so sharp and thick it can put someone's eye out, and far too thick to cut with scissors. I had to resort to multiple passes with an X-Acto(TM) Knife.

On the other hand the approximation is very simple. 180/13 = 13.8 degrees and tan(180/13) = 0.246. Arctan (1/4) = 14.0 degrees, an error of only 0.2 degrees. Can we really pass this up?

Start with a square sheet folded diagonally in half.

Fold the two bottom corners together

Unfold the paper.

Fold the paper lengthwise in half

Unfold the paper.

Fold the base of the paper up to the horizontal center crease

Unfold the paper.

Fold the paper along the diagonal of the lower right eighth. You will definitely want a straightedge to fold the paper against.

Now unfold the paper to reveal the crease

Fold the left side over so the base lines up with the diagonal crease. For reasons that will become clear, we want to keep that lower wedge exposed. The angle of the gray wedge is (6/13)180.

Bisect the gray wedge. The angle of the gray wedge is now (3/13)180. What saves us from having to trisect an angle is that the overall angle, counting the original exposed corner, is (4/13)180.

Here's the angle bisected. At this point you will probably find the paper getting unwieldly and you may want to undo some folds and re-crease them more sharply.

And one last time. Realistically you will probably want to undo the prior folds and bisect them one at a time.

Cut the wedge as desired. With stars with lots of points, there's a good chance the long dimension of the wedge will be about equal to a diagonal radius of the original square. You'll need something a lot more robust than scissors for this. Unfold the paper as shown below.

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Created 22 March 2006, Last Update20 January 2020