Convex Tangram Polygons

If you’re a regular reader of this blog, or visitor to my website, you probably know of my long-standing interest in geometric puzzles. Among those, tangrams are probably the most well-known and widely available. Thus I included them in Geometry Labs (free download) where they are the subject of Section 2, and Lab 10.6. In this post, I will highlight a couple of tangram challenges from the book, and add another one to that list.

From Lab 2.3: Explain why a six-piece square is impossible. If you are not having your students work on this lab, a better way to present this is as follows:

Is it possible to make a square, using 1, 2, 3, …, 7 tangram pieces? If yes, record your solution. If not, prove that it is impossible.

From Lab 2.5: How many seven-piece convex tangram figures can you find? As it turns out, there are only 13 such figures. Finding them all would make a good whole-class challenge. I had not included them in the Geometry Labs answers, so I corrected this omission in this PDF.

Extension to Lab 2.5: For what n can you make a convex tangram n-gon? (you may use any number of pieces)

A convex triangle or quadrilateral can each be made with a single piece, or with multiple pieces. It is not too difficult to make a convex pentagon or hexagon. A convex seven-gon is more difficult to find, and cannot be made with seven pieces. Here is a six-piece solution:


One way to think about this problem is to pay attention to the exterior angles. An ant walking around the perimeter of a convex figure would turn a total of 360° at the vertices. To get more vertices, the exterior angles need to be as small as possible, and the interior angles need to be as large as possible. But all tangram angles are multiple of 45°. The greatest possible interior angle is 135°, since the next bigger one would be 180°. It follows that the least possible exterior angle is 45°. But 8 ⨉ 45° = 360°, so a convex nine-gon is definitely impossible.

In the seven-gon above, all but one of the interior angles are 135°. To make a convex eight-gon would require all interior angles to be 135°. As it turns out, the only way to achieve that is to have a hole, which of course defeats our purpose:


We conclude that the only convex tangram n-gons are for values of n ranging from 3 to 7, inclusive.

I’ll add this extension to the Geometry Labs page, where I keep track of “Connections, Corrections, Extensions, and Revisions” — many of them contributed by other teachers.

— Henri

PS: I used the Virtual Tangrams applet on my website to make the figures in this post.

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