Seven Ways to Fold a Regular Pentagon

This page has links to various methods for folding a square to construct a regular pentagon. An overview and comparison of the methods is also provided here.

Overview

After folding, the paper is generally cut so that the cut edges are the edges of a regular pentagon. Alternatively, once the edges of the pentagon are defined, the paper can be folded on those edges to form the pentagon. Then, the pentagon-based model can truly be said to be “folded from a square”, but the thickness of the model will generally be uneven. But both squares and regular pentagons are regular polygons, and the only reason that we don’t construct squares from pentagons is that paper is readily available in squares. We may use both folding and cutting to make a pentagon, but most models made from pentagons are folded from a pentagon without cutting.

The various methods for constructing a regular pentagon from a square differ in complexity and accuracy. The accuracy of the finished pentagon is partly due to the careful precision of the folder, and partly due to the mathematical accuracy of the geometry of the construction method. Some methods have no (or negligible) geometric error, but due to the complexity, the human errors accumulate even with careful folding. Other methods have some geometric error, but due to the simplicity of the folding, the human errors do not accumulate; and sometimes the geometric error can be practically removed by a simple adjustment. You can test whether you have made a perfect pentagon by folding it in half five different ways (folded edge goes from a corner to the midpoint of the opposite side). For each way, the two halves should match exactly.

Comparison of Methods

Below, the length of one side of the pentagon is given, scaled for one side of the square = 1.

Method 1 is my modification of a method published in the “Origami Basics” section of the Origami USA Convention books and credited to Alice Gray. I think it is the easiest one to fold. One of the landmarks, however, provides an approximation for a basic angle, and causes a small geometric error in the pentagon construction (one tenth of an inch for a 10 1/2 inch square). However, I provide a guide for shifting the landmark to correct the error. Also, in step 9 of the OUSA diagrams, the pentagon is made smaller than it needs to be. My modification of the method makes a larger pentagon. The only creases on the pentagon are those that divide the pentagon in half. Pentagon side = 0.55047

Method 2 is my modification of a method published in the Origami USA May 1987 Convention book as part of the instructions for David Shall’s Five Pointed Star on page 108. I like this one best. It has a geometric error that is correctable, but so small that it is hardly worth trying to correct (less than one 50th of an inch for an 11-inch square). It is nearly as simple as method 1, and makes an 8.1% larger pentagon. It can be folded so that nearly the only creases on the pentagon are those that divide the pentagon in half. The center of the pentagon is at the center of the square (not so for the other methods). Pentagon side = 0.59511

Method 3 is a construction that I developed that makes the largest possible regular pentagon from a square. The geometric error is practically zero (less than 50 parts per million), but due to development of the landmark in stages, human errors easily accumulate. Also, many extra creases are left on the paper that do not relate to the symmetry of the pentagon. It is an interesting construction, but the extra difficulty and poorer quality isn’t worth getting a pentagon that is only 1.2% larger than method 4. Pentagon side = 0.62574

Method 4 is my modification (to reduce a geometric error) of a method credited to Fred Rohm that I recently learned. Unmodified, it has a geometric error about twice as large as for uncorrected method 1, but my modification makes the geometric error so small that human error is most of the error. It makes a 3.85% larger pentagon than method 2, but leaves a few extra creases on the paper that do not relate to the symmetry of the pentagon. With a little practice, the extra creases might be kept mostly off the pentagon, so if pentagon size is important or if extra creases are unimportant, this method might be considered better than method 2. Pentagon side = 0.61803 (the inverse of the Golden Ratio)

Method 5, by David Chandler, has no geometric error, and makes a pentagon the same size as for method 4. But due to the complexity, the human errors accumulate; and due to more paper layers on folds, more physical errors occur. An experimental comparison with method 4 and measurements indicated about eight times more error than method 4 in practice. Also, extra creases are left on the paper that do not relate to the symmetry of the pentagon. Pentagon side = 0.61803 (the inverse of the Golden Ratio)

Method 6, by David Dureisseix, has no geometric error, makes the largest possible regular pentagon, and is simpler than Method 3, thus having less human error.  The method actually makes a "stellated pentagon" (a 5-pointed star), but a true pentagon can be made by folding lines from one star-point to the next.   The article Folding the Regular Pentagon by Bisections and Perpendiculars by Roger C. Alperin presents a mathematical analysis of the method.  Pentagon side = 0.62574

Method 7, suggested to me by Ralph Jones, also makes the largest possible regular pentagon.  I made the diagrams from his description, and added an adjustment that eliminates nearly all of the geometric error.  Ralph Jones also suggests using a gauge sheet to position the landmarks.  Ralph has provided a gauge sheet that you can print, and an explanation of how to use it.  Pentagon side = 0.62574

Overall, I think method 2 is best, with method 1 nearly as good. With the modifications, and careful folding, methods 1, 2, and 4 can be used with errors less than the thickness of 2 to 4 paper layers.