Tangram Solver
Tap the little triangles to build your silhouette. Each square of the board splits into four of them along its diagonals, which is what lets a tan sitting at an angle be described exactly.
The seven tans come to sixty four of those triangles between them, so the Solve button only wakes up when your shape is exactly that size. It is the quickest way to know a shape is hopeless before you start: if the area is wrong, nothing else matters.
What it actually does
It tries to cover your shape with the seven tans, every piece used once, none overlapping. Rather than sliding shapes about and testing for collisions, it works on the quarter triangles: each tan in each position becomes a set of them, and covering the shape exactly once is a problem with a name, exact cover, and a search that is good at it.
That is also why the answer comes back quickly on shapes that work and slowly on shapes that nearly work. A silhouette with one triangle in the wrong place forces the search to try almost everything before it can rule it out.
When it says no
Two different things can go wrong and the tool tells you which. No arrangement makes that shape means the search finished and ruled every possibility out, which is a proof. Gave up means it hit its own limit first, so the shape might still be possible and the tool simply could not settle it.
Most refusals are the first kind, and most of those are area. Sixty four quarter triangles is not a guideline: a shape one triangle short cannot be made by these seven pieces however clever you are about it.
How to know it is right
The oldest fact about this puzzle is that the seven tans came from a square, so they must tile that square. Load The square it comes from and press Solve: if that fails, nothing else the tool says is worth anything. It is the first test this solver had to pass and it is the one worth repeating whenever the piece shapes are touched.
There is a fuller history in the tangram write-up, and a daily tangram on this site with a new silhouette to fill each day.
For the same instinct in wood, where the pieces interlock rather than lie flat, see the Fish Tank and the rest of the 2D packing puzzles.
Frequently Asked Questions
How many pieces does a tangram have?
Seven: five triangles in three sizes, one square and one parallelogram, all cut from a single square.
Can a tangram make any shape?
No. The seven pieces have a fixed total area and all their edges meet at 45 degree angles, so a silhouette has to be the right size and the right geometry before it can be covered at all.
Is the tangram parallelogram allowed to be flipped?
Yes, and it is the only piece where flipping matters, because it is the only one that is not its own mirror image.