After my adventures figuring out every Tangram square and parallelogram, I had to complete the trilogy with the triangle, didn’t I? I spent hours agonising over whether I had found every parallelogram, but I couldn’t stop there. As a proud completist, I present you with every Tangram triangle.
Five of the seven Tangram pieces are triangles. In a sense, it all starts with the triangle. The other two pieces – the square and the parallelogram – can be built by combining the two small triangles. This final part of the trilogy focuses on the native geometry of the Tangram set.
If I thought building triangles from the subset of pieces would be the easiest, most effortless part of the entire project, I was wrong. Discovering them was far from trivial, and contained an intriguing surprise.
Definitions
All five triangles in the Tangram set are isosceles right-angled triangles.
The word isosceles comes from the Ancient Greek word ἰσοσκελής (isoskelēs). Translated literally, it means “equal-legged.”
iso- meaning “equal” or “same” (the same root in isotope or isometric).
–sceles meaning “leg” (or the side of a geometric figure).
While isosceles tells you about the sides (“equal legs”), right tells you about the angle:
Right-angled: Each triangle features one strict 90 degree corner.
The two shorter sides that meet at that right angle are identical in length, which means their base angles are both 45 degrees.
Combining both gives us isosceles right-angled triangle: a triangle with two equal legs that meet upright at a perfect 90 degree corner.
Why is Isosceles so hard to spell? Thankfully, I don’t think I’ll need to type it as often as I did parallelogram.
24 Triangle Solutions
Definitions over, here are the solutions, leaving out simple rotations and reflections (across an imaginary mirror line outside of the shape). I have also left out images where I could have swapped a small orange triangle for a yellow one, or exchanged the large blue triangle with the pink one: these are just colour swaps, not a new configuration of the pieces.
1 piece
2 pieces
3 pieces
4 pieces
5 pieces
Remember the 7-piece square? Peeling away the two large triangles gives us this 5-piece triangle.
6 pieces
Just like with the square, no solution is possible. See below for an explanation.
7 pieces
For those of you who like their data summarised:
| Number of pieces | Number of solutions |
| 1 | 3 |
| 2 | 2 |
| 3 | 3 |
| 4 | 9 |
| 5 | 1 |
| 6 | 0 |
| 7 | 6 |
| Total | 24 |
Look at the distribution: instead of a smooth Bell Curve, there is a massive spike in the middle, followed by a sudden drop-off, then ending with another spike. The distribution of the triangle family is unique compared to the squares and parallelograms.
Why can’t you make a triangle with 6 pieces?
Like the square, there are no solutions with 6 pieces.
When you attempt to build any shape using 6 pieces, you have to leave out one piece. Depending on which piece you choose to omit, the 6 remaining pieces will always add up to one of three specific totals:
Leave out a Small Triangle (1 unit): Your remaining 6 pieces total an area of 15 units.
Leave out a Medium Triangle, Square or Parallelogram (2 units): Your remaining 6 pieces total an area of 14 units.
Leave out a Large Triangle (4 units): Your remaining 6 pieces total an area of 12 units.
So, any 6-piece shape must have a total area of either 12, 14 or 15 units.
An isosceles right-angled triangle requires strict proportional harmony between its legs and its hypotenuse. When you attempt to arrange 6 pieces totaling an area of 12, 14 or 15 units into a single, unified isosceles right-angled triangle:
The angular constraints do not line up.
The lengths of the combined outer edges (hypotenuses versus legs) refuse to match up symmetrically.
The geometry of those specific subsets (12, 14 or 15 units) cannot form a clean 45 degree/45 degree/90 degree triangle.
The shared impossibility of making a square or triangle with 6 pieces is one of the most satisfying mathematical symmetries in this trilogy. The ability to make five different parallelograms with 6 pieces adds in an asymmetry that I refuse to find annoying.
Breaking Binary Scaling: The Area Surprise
While triangles are similar to squares in their refusal to have 6-piece solutions, there is a way in which triangles are much more flexible than squares: they successfully break binary scaling.
Using 4 pieces, there are nine solutions:
Perhaps you think they are all the same size. Look closer. Or, better still, make them with your own Tangram sets. Surely you’ve bought some by now.
The solutions in the second row are bigger than those in the first row.
The first row solutions all add up to an area of 8 units:
Solutions 1 and 2: 1 Large Triangle + 1 Medium Triangle + 2 Small Triangles = 4 + 2 + 1 + 1 = 8
Solutions 3 and 4: 1 Large Triangle + 1 Parallelogram + 2 Small Triangles = 4 + 2 + 1 + 1 = 8
Solution 5: 1 Large Triangle + 1 Square + 2 Small Triangles = 4 + 2 + 1 + 1 = 8
The second row solutions all add up to an area of 9 units:
Solution 1: 1 Large Triangle + 1 Square + 1 Parallelogram + 1 Small Triangle = 4 + 2 + 2 + 1 = 9
Solution 2: 1 Large Triangle + 1 Square + 1 Medium Triangle + 1 Small Triangle = 4 + 2 + 2 + 1 = 9
Solutions 3 and 4: 1 Large Triangle + 1 Parallelogram + 1 Medium Triangle + 1 Small Triangle = 4 + 2 + 2 + 1 = 9
When building the squares, every single valid area was strictly locked to powers of two (2, 4, 8, 16) because of how symmetric scaling works.
Triangles don’t require that strict binary scaling symmetry. You can mix and match pieces in combinations that add up to 9, an odd integer. Unlocking an area of 9 proves that triangles don’t follow the strict binary area scaling that locked the squares down.
Stumbling onto an odd, non-binary area like 9 was an unexpected surprise after hitting the rigid limits of the squares.
The Trilogy’s Evolution of Area
Squares are the Binary Fortress: restricted strictly to powers of two – 2, 4, 8, 16 – every area solution must double precisely.
Parallelograms are the Even-Numbered Step: expanding into 2, 4, 6, 8, 12, 16 area solutions, they break free from the powers of two trap by allowing even multiples, but they still stubbornly shut out odd numbers.
Triangles are the Liberated Frontier: they possess the geometric freedom of 1, 2, 4, 8, 9, 16 area solutions. They shatter the even-number rule to pull off an unexpected odd-area construction of 9.
Orientations
Did you notice the different orientation of the large blue triangle?
The First Row (5 solutions) contains the Vertical Spine Family. In these builds, the large pale blue triangle stands upright, using one of its vertical legs as an internal dividing wall. Its hypotenuse forms the outer slanted slope on the left side rather than running down the centre.
The Second Row (4 solutions) contains the Base Foundation Family. In these solutions, the large pale blue triangle shifts orientation entirely, lying flat along the baseline with its hypotenuse resting squarely against the ground, forcing the remaining pieces to stack diagonally above its upward-pointing apex.
Every possible Tangram triangle
For the sake of completeness, here they are in one image. Beginning as a slender spire of 1, 2, and 3-piece builds, the layout quickly fans out into a striking geometric hourglass. Note all those 4-piece variations, the tight pinch through a solitary 5-piece solution, before opening once more into the broad foundation of the 7-piece triangles.
Self-similarity
Every individual triangle is an isosceles right-angled triangle. Every solution is an isosceles right-angled triangle.
This is a beautiful display of self-similarity in geometry. Whether it is a simple 1-piece triangle or the array of 7-piece solutions at the base, the angles and proportions remain locked in harmony.
How the Squares, Parallelograms and Triangles Compare
The number of solutions for each shape:
| Number of pieces | Squares | Parallelograms | Triangles |
| 1 | 1 | 1 | 3 |
| 2 | 2 | 2 | 2 |
| 3 | 1 | 5 | 3 |
| 4 | 5 | 10 | 9 |
| 5 | 2 | 6 | 1 |
| 6 | 0 | 5 | 0 |
| 7 | 2 | 2 | 6 |
| Total | 13 | 31 | 24 |
The Geometric “Sweet Spot”
For every shape, it is always some combination of 4 pieces that gives the most solutions.
When building shapes with Tangram pieces, you face two opposite limitations at the extremes:
1 to 3 pieces: you simply don’t have enough pieces available to construct many different arrangements.
5 to 7 pieces: you usually have to include the two large triangles which forces the layout into rigid, pre-determined arrangements with less room for variation.
4 pieces is the exact balance point. You can combine small, medium and large pieces to experiment with different internal arrangements, while the total area (6, 8 or 9 units) is large enough to allow multiple distinct piece combinations.
Project completion
From the powers of two maths that explains why it is impossible to make a square with 6 pieces, to the chirality and aspect ratios of the parallelograms, to the triangle solutions with areas of 9 units, it’s been an eventful, if somewhat angular, journey.
I’m not going to claim originality where it might not exist. At least one other person must have played about with their Tangram set and explored whether they could make different sized squares, parallelograms and triangles using a subset of the pieces. If I have created original recreational mathematics content, that’s a bonus. What’s most important is that it was fun and I learnt a lot. That’s a winning combination in any area of life.
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