Tangram Puzzle and Piece Reference
The area test rejects an impossible silhouette in one pass and can never confirm a possible one. Two shapes here prove it.
Right area, wrong shape
The seven pieces have areas 1, 1, 2, 2, 2, 4 and 4 in units where the square is 16. So there is a test for whether a silhouette can be made that costs one pass round its outline: if the area is not exactly 16, no arrangement exists and no search is needed. Two of the shapes in the menu here are rejected that way before anything is tried.
What the test cannot do is the other direction. Of the 5 silhouettes here, 3 have area exactly 16, and 2 of those still have no arrangement — the 8 by 2 rectangle and the house. Right area, wrong shape. An area check is a cheap way to say no and no way at all to say yes, which is worth knowing before you spend twenty minutes on a silhouette somebody drew by eye.
That distinction bit while this page was being written. An earlier version of the piece set passed an area check and a full tiling check — forty thousand sample points, none uncovered, none double-covered — and was still wrong: one "parallelogram" had two collinear edges, so it was really a triangle with a redundant corner, and the set was six triangles and a square. Both checks look at aggregates. Neither looks at what the pieces are.
Only one piece has a left and a right
Six of the seven pieces look identical in a mirror — turn one over and no rotation is needed to put it back. The parallelogram is the exception. Its mirror image cannot be reached by any rotation at all, which is why a physical tangram set sometimes has to be flipped, and why that one piece is often printed a different colour on each face.
It is easy to check this the wrong way. Compare the cyclic sequence of side lengths and turn angles — a genuine rotation invariant — and the parallelogram matches its own mirror, because that sequence is palindromic. It reports as symmetric and is not. The only reliable test is to actually rotate the mirror image and look for a match, which is what this page does.
The seven pieces
Areas in units where the whole square is 16. The legs step by a factor of root two.
| Piece | Shape | Area | Sides | Mirror |
|---|---|---|---|---|
| Large triangle | right isosceles triangle | 4 | 4.00, 2.83 | same either way |
| Large triangle | right isosceles triangle | 4 | 4.00, 2.83 | same either way |
| Medium triangle | right isosceles triangle | 2 | 2.00, 2.83 | same either way |
| Square | square | 2 | 1.41 | same either way |
| Parallelogram | parallelogram (not a rectangle) | 2 | 1.41, 2.00 | differs — chiral |
| Small triangle | right isosceles triangle | 1 | 2.00, 1.41 | same either way |
| Small triangle | right isosceles triangle | 1 | 2.00, 1.41 | same either way |
Two small triangles make a medium, and two mediums make a large — the legs are in the ratio 1 : √2 : 2. That is also why two small triangles can stand in for the square or the parallelogram in some figures but not others: the areas match, and the shapes do not.
How to use
- Pick a silhouette; the area verdict appears immediately.
- Click a piece to select it, then move it with the arrow keys.
- Press R to rotate by 45 degrees and F to flip.
- Only the parallelogram is changed by flipping — the rest are symmetric.
Frequently asked questions
What are the seven tangram pieces?
Five right isosceles triangles in three sizes — two large, one medium, two small — plus one square and one parallelogram. In units where the whole square is sixteen, their areas are 4, 4, 2, 1, 1, 2 and 2. The triangle legs step by a factor of root two, which is why two small triangles make a medium and two mediums make a large.
How can I tell if a shape is impossible?
Measure its area. The seven pieces come to exactly sixteen units, so any silhouette with a different area cannot be made whatever you try, and establishing that takes one pass round the outline. It is the cheapest useful thing you can know about a tangram figure.
Does the right area mean a shape can be made?
No, and this is the part people skip. An area check is a cheap way to say no and no way at all to say yes. Two of the silhouettes here have area exactly sixteen and still have no arrangement — an eight by two rectangle and a house shape. Right area, wrong shape. Necessary is not sufficient.
Why is the parallelogram special?
Because it is the only chiral piece. Its mirror image cannot be reached by any rotation, so a physical set has to be turned over to make certain figures. Every other piece looks identical in a mirror, which is why the parallelogram is often printed a different colour on each face while the rest are not.
How do you test whether a shape is chiral?
By actually rotating the mirror image and looking for a match. The obvious shortcut fails: comparing the cyclic sequence of side lengths and turn angles is a genuine rotation invariant, but a parallelogram's sequence is palindromic, so it matches its own mirror and reports as symmetric. That shortcut says all seven pieces are achiral, and it is wrong about exactly one.
How many shapes can a tangram make?
Endlessly many in general, but only thirteen convex ones — a result proved by Fu Traing Wang and Chuan-Chih Hsiung in 1942. That figure is often quoted without the word convex, which makes it sound absurdly restrictive; the interesting fact is not that there are thirteen shapes but that the convex ones are so few.
Where does the tangram come from?
China, some time before 1800 — the earliest known references are from the first years of the nineteenth century, and it reached Europe and America around 1817 where it became a genuine craze. The name is not Chinese; in Chinese it is qi qiao ban, roughly "seven boards of skill".
Can two small triangles replace the square?
They have the same area — two units between them against the square's two — but not the same shape, so only sometimes. Two small triangles joined along their hypotenuses make a square of the right size, so in that arrangement yes; joined along a leg they make a larger triangle instead, which does not fit where the square goes. Equal area is not interchangeability, which is the same lesson as the silhouette test.
Why do the pieces have those particular sizes?
They all come from repeatedly halving the square along a diagonal. Cut the square in two, cut one half in two again, and so on — every piece is a power-of-two fraction of the whole, which is why the areas are 1, 1, 2, 2, 2, 4, 4 and why the triangle legs are in the ratio 1 to root two to two.
Is there a systematic way to solve a silhouette?
Place the two large triangles first, since between them they are half the total area and there are usually very few positions where they both fit. After that the medium triangle, then the square and parallelogram, leaving the two small triangles to fill whatever is left — they are the most flexible and therefore the least informative to place early.
Why are the pieces here on a grid?
Because every tangram vertex lands on a lattice point when the square is drawn four units across, and every edge runs horizontally, vertically or at forty-five degrees. That is what makes the dissection checkable by computer: the whole square splits into sixty-four quarter-triangles, and each piece is an exact union of them.
Does this store anything?
Nothing. The puzzle runs entirely in your browser, there is no upload, and nothing is saved between visits. There is no global board on this one either — a silhouette has no natural score, and timing a free-form arrangement would rank fiddling rather than solving.
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