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.
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