Completing the Tangram Trilogy: Every Triangle and an Area Surprise

On 14/10/202614/10/2026 By Games4LifeIn Recreational Maths1 Comment

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:

Tangram triangle using 4 pieces and 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:

Tangram triangle using 4 pieces and 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.

Did you enjoy this guide? If it inspired you to show some children you know that there is more to the Tangram than cats, boats and walkers leaning into the wind, consider leaving a tip for Games4Life to help support future content.

Slanted Logic: Solving Every Unique Tangram Parallelogram

On 23/09/202614/10/2026 By Games4LifeIn Recreational Maths1 Comment

If you landed here without first reading my blog on every Tangram square, I recommend you start there. This work flows naturally from that.

Having solved every Tangram square, the next logical step was to solve every Tangram parallelogram. My motivation was the same: I searched online and could not find every parallelogram using a subset of the Tangram pieces.

Definitions

Always a good place to start. A parallelogram is a type of quadrilateral. It has two pairs of parallel sides; the opposite sides are of equal length; the opposite angles are of equal measure.

The parallelgram supplied in the Tangram set is an Oblique one: its adjacent sides are of unequal lengths and the angles are not right angles. The informal name is the slanted or leaning parallelogram.

We need to distinguish the Oblique parallelogram from these special cases of parallelograms:

Square: a parallelogram with four sides of equal length and four right angles.

Rectangle: a parallelogram with four right angles.

Rhombus: a parallelogram with four sides of equal length.

Math textbooks will tell you there are only 3 parallelograms made with all 7 pieces. In 1942, mathematicians Fu Traing Wang and Chuan-Chih Hsiung published a famous paper proving that using all 7 pieces, you can form exactly 13 convex shapes – 3 of which are parallelograms (the Square, the Rectangle, and the Slanted Parallelogram). But if you include every subset from 1 piece up to 7 pieces, there are 31 distinct parallelograms hiding in your Tangram set.

31 Parallelogram Solutions

Definitions and explanations 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

6 pieces

7 pieces

For those of you who like their data summarised:

Number of pieces	Number of solutions
1	1
2	2
3	5
4	10
5	6
6	5
7	2
Total	31

Solutions with 6 pieces

In my post on every Tangram square, I demonstrated why there are no squares with 6 pieces. But the images above show you can make a parallelogram with 6 pieces. In fact, you can make five different parallelograms with 6 pieces.

To understand why, let me quickly recap how piece areas work. We can express the size of each piece as a unit of area:

The small triangle = 1 unit

The square = 2 units

The parallelogram = 2 units

The medium triangle = 2 units

The large triangle = 4 units

Part of what defines a square is that all four sides must be equal. Parallelograms have no such restriction: the side lengths do not need to be equal. If a square is like a drumhead, stretched tight over a fixed frame, a parallelogram is like an accordion: you can squish the height down and stretch the base wide.

Because the sides don’t have to be equal, an area of 12 units cleanly breaks down into a simple height of 2 and a base of 3. Since a 2 x 3 shape has an area of 6 grid squares – which cleanly equals 12 small-triangle units of area – and building lengths of 2 and 3 using Tangram legs is effortless, the shape snaps together seamlessly. You can check this for yourself by looking at the 6-piece images above: every solution sits 2 tall, stretches 3 wide, and covers an area of 12 units!

The simple geometric freedom the parallelogram possesses – its side lengths not needing to be equal – is why 6-piece parallelograms come in five unique solutions, while 6-piece squares are impossible.

Chirality

Of all the pieces in the Tangram set, the parallelogram is the only one that is chiral. Chirality (from the Greek cheir, meaning “hand”) means an object has built-in handedness.

Humans hands and feet are chiral. Hold your hands in front of you with both palms facing forward. Your left thumb points right, and your right thumb points left. No matter how you spin your hands in the air, you can’t get both palms facing forward with both thumbs pointing the same way. That is why a left glove won’t fit a right hand unless you turn it inside out. No matter how you rotate the parallelogram, you can never get it to line up exactly on top of its mirror twin.

When I watched the children in my Curious Minds Club make every Tangram square, they realised they sometimes needed to flip the parallelogram over to make a different solution. If they rotated it on the table they could never get it to fit. They worked out that the only way to turn a left-leaning parallelogram into a right-leaning parallelogram was to lift it off the table into three-dimensional space and flip it over. Being present at those lightbulb moments is why working with children is so rewarding. In the Tangram set, the parallelogram is the only piece needing a flip. The square and triangles can be rotated to make their mirror image.

A Tale of Two Ratios: Ribbon versus Chunky Parallelograms

When you look at the 3-piece parallelograms, you hit a fascinating geometric quirk: all five solutions share the exact same area, yet they look markedly different.

Every one of these shapes uses a total area equal to 4 small triangle units. But depending on how you arrange the internal pieces, the shape stretches or compresses like an accordion.

We can group all five 3-piece solutions into two distinct “families” of aspect ratios:

1. The “Ribbon” Family (4 : 1 Aspect Ratio)

In the first pair of solutions, the pieces align flat along their hypotenuse orientation. The base is four times the height, giving an aspect ratio of 4 : 1. Because the height is squished down, the shape has to stretch out to a sweeping length to fit all its area. The result is a long, thin, ribbon-like silhouette.

2. The “Chunky” Family (2 : 1 Aspect Ratio)

In the other three solutions, the pieces rotate onto their legs at 45 degrees. The base is now just twice the height, giving an aspect ratio of 2 : 1. To maintain the identical surface area of 4 small triangle units, the base contracts, cutting the aspect ratio clean in half into a stockier, taller shape that is exactly twice as wide as it is tall.

📏 The 1-Minute Lab Test: Proving the Maths with a Ruler

I decided not to take the theoretical maths on faith. I grabbed my ruler and measured the wooden pieces.

The Ribbon Parallelogram: Measures 92 mm long and 23 mm high.

The Chunky Parallelogram: Measures 64 mm long and 32 mm high.

A reassuring 4 : 1 versus 2 : 1 split in real life! My physical measurements were right on the money once we account for real-world wooden bevels and manufacturing margins.

Every possible Tangram parallelogram

For the sake of completeness, here they are in one image. Is it a giant Tangram Christmas tree? On top sits a single red parallelogram, a sleek, minimalist star topping a glowing fir tree of pure geometry.

13 and 31

I discovered 13 Tangram squares and 31 Tangram parallelograms. I love that both of these are prime numbers, and are the inverse of each other. Beyond just being prime, 13 and 31 belong to a rare class of prime numbers known as emirps (“prime” spelled backwards). An emirp is a prime number that results in a different prime number when its digits are reversed (palindromic primes like 101 don’t count). 13 and 31 are the first two-digit emirp pair in base-10 mathematics.

The jump from 13 to 31 represents the difference between strict symmetry and geometric freedom:

Squares (13 in total): A square is hyper-constrained. All four sides must be identical in length, all four angles must be 90 degrees, and its side length is strictly locked to the square root of the area. These constraints completely eliminate 6-piece squares.

Parallelograms (31 in total): A parallelogram relaxes both constraints. Opposite sides must be parallel, but adjacent sides can be unequal, and interior angles can combine 45 and 135 degrees. This extra flexibility lets you adjust the aspect ratio, unlocking 6-piece combinations and more than doubling the total count.

While geometry dictates the shapes, base-10 mathematics gives us a poetic numerical twist. I didn’t expect to uncover a coincidental link between two different branches of mathematics – recreational geometry and number theory – but that’s the beauty of playing about with shapes on your lounge carpet.

Bonus Round: The Hollow Parallelogram (A Parallelogram inside a Parallelogram)

Think we’ve exhausted all 7 pieces? Think again.

If we shift our thinking from building solid shapes to building a frame, all 7 pieces lock together to form a hollow parallelogram, with a perfectly matching parallelogram void right in the centre.

I’ve relaxed my rule of not showing rotations as this is so special: the second image is a 180-degree rotation of the first.

To calculate the total boundary, we add together the combined area of the 7 pieces (16 units) with the hole (2 units) to give an effective area of 18 units.

Tribulations

Finding all 31 solutions took hours of trial and error, and double-checking rotations and mirror images. Several times I thought I had finished, but when I looked closer I realised there was one more to discover. As well as working systematically through the number of different pieces, the key was looking at the way the pieces fit together in the 3-piece solutions, working out where an extra piece could fit to make the 4-piece solutions, then extrapolating this method to a higher number of pieces.

This blog is my best endeavour to find every Tangram parallelogram. In recreational maths, the crowd is always smarter than the individual. Can you find a 32nd solution? If you build a valid Tangram parallelogram that isn’t in my master photo, send me a picture and I’ll add it to the Wall of Fame!

I’ll end by stating that I hope I don’t have to type parallelogram again for a long time.

Ready for the next shape? Check out how I solved all 24 Tangram triangles.

Did you enjoy this guide? If it gave you ideas for an activity to do in your school or filled a gap in your puzzle skillset, consider leaving a tip for Games4Life to help support future content.

I Solved Every Single Tangram Square (So You Don’t Have To)

On 14/09/202623/09/2026 By Games4LifeIn Recreational Maths2 Comments

I bought some wooden Tangram sets to use at my Curious Minds after school club. Inside a wooden frame fit seven differently-coloured pieces: two large triangles; one medium triangle; two small triangles; a square; and a parallelogram. The frame is a square, and all seven pieces fit together to fill the space with no gaps and no overlaps.

“No gaps and no overlaps” is the univeral requirement when tessellating shapes. A tangram is not a tessellation, but the two concepts are closely related in geometry. It seems plausible that the study of tessellations helped inspire the creation of geometric puzzles like the tangram. My blog on two dimensional tessellations has turned out to be one of my most popular posts.

Playing around with the seven tangram shapes, I quickly saw it was possible to take a subset of the seven pieces and make different sized squares. Despite my best efforts, I could not find a website, blog post or activity sheet which showed every possible square. There are a wealth of resources for animal, bird or flower outlines, plus plenty of exotic shapes. As much as I was tempted to solve Duck in a Hurry and the Contented Rabbit, there was nothing for the humble square. The challenge was obvious: to create my own post to fill this gap for others.

I spent a happy hour sitting on the floor figuring out every combination. Here they are, 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.

13 Square Solutions

1 piece

2 pieces

3 pieces

4 pieces

Tangram square using 4 pieces. 1 large triangle, 1 medium triangle and 2 small triangles.

5 pieces

6 pieces

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	1
2	2
3	1
4	5
5	2
6	0
7	2
Total	13

If you are tempted to spot a pattern between the number of pieces and number of solutions, I didn’t get anywhere either!

Why can’t you make a square with 6 pieces?

For each piece, we can express its size as a unit of area:

The small triangle = 1 unit

The square = 2 units

The parallelogram = 2 units

The medium triangle = 2 units

The large triangle = 4 units

For any square, the total area must be either 2, 4, 8 or 16 units. Did you notice these are the successive powers of 2? That’s not a coincedence. Every time you step up to the next allowed square size, you are exactly doubling the area of the previous square. This is scaling symmetry, a beautiful, repeating geometric pattern. I will explore this further in a future blog.

The biggest square we can make with all 7 pieces has a total area of 16 units (1 + 1 + 2 + 2 + 2 + 4 + 4).

The squares with 5 pieces have a total area of 8 units (1 + 1 + 2 + 2 + 2).

The squares with 4 pieces have a total area of 8 units (1 + 1 + 2 + 4). (Note that any of the 2-unit pieces work).

The squares with 3 pieces have a total area of 4 units (1 + 1 + 2).

The squares with 2 pieces have a total area of either 2 units (1 + 1) or 8 units (4 + 4).

The square with 1 piece has a total area of 2 units (2, or the square by itself).

For any combination of 6 pieces you choose, there is no way to make a square with either 2, 4, 8 or 16 units. Lets look at what happens if you try to make a square using exactly 6 pieces. You have to leave one piece behind, which ruins the required “power of 2” area:

Leave out a large triangle (4 units): Remaining area = 16 – 4 = 12 units.

Leave out a medium triangle, square or parallelogram (each 2 units): Remaining area = 16 – 2 = 14 units.

Leave out a small triangle (1 unit): Remaining area = 16 – 1 = 15 units.

Because areas of 12, 14, and 15 cannot geometrically form a square using the standard fixed angles of a tangram set, a 6-piece square is mathematically impossible! Play about with 6 pieces so you can see how they stubbornly refuse to fit into a square.

Is one of the 4-piece solutions a reflection of itself?

Someone asked me if the solution on the right is a reflection of the one on the left.

I stated above that I am leaving out simple rotations and reflections. To get from the left solution to the right solution requires a combination of reflecting and rotating, so I categorise them as two different solutions. You can either: reflect the left solution clockwise by 90 degrees, then reflect it; or, you can reflect the left solution, then rotate it anti-clockwise by 90 degrees. Either way, something interesting happens: the yellow and orange triangles swap places! Try it yourself and see if you can figure out what is going on.

Every possible Tangram square

For the sake of completeness, here they are in one image. Does it look like a giant tangram person? But which way is it facing?

When I took my tangram sets to my after school Curious Minds club they were very popular. I had three boys at that session. We got into a beautiful groove. For example, I told them to take the purple piece, the yellow piece and the orange piece and make a square bigger than the one before. They were really good at focussing on their own pieces and not sneaking a look to see someone else’s solution. As we progressed to 5 and 7 pieces they did need some assistance, but not as much as I expected. By this stage, they had developed a feel for how the pieces fit together. It was helpful that with 7 pieces they got to make the square inside the wooden frame, which naturally guided them to the total area of the solution.

Feel free to contact me if you think I successfully filled a gap in your tangram knowledge.

Ready for the next shape? Check out how I solved all 31 Tangram parallelograms.

Did you enjoy this guide? If it saved you time or filled a gap in your puzzle skillset, consider leaving a tip for Games4Life to help support future content.