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Tangrams: pictures from seven pieces

Geometry · level 1, lesson 5 of 5

A question to start

Take a square. Cut it into just seven pieces. Now, using those same seven pieces, make a cat. Then a boat. Then a running person, a house, a bird, a rabbit, a candle, a teapot.

How many different pictures do you think you can make from the same seven pieces? Ten? A hundred?

People have been playing this game for more than two hundred years, and they have made thousands of different pictures. The puzzle is called a tangram, and it comes from China. In this lesson you will find out what the pieces are, how they fit together, and a secret about their sizes that makes every tangram picture work.

The idea

The seven pieces

Here is a tangram square, with all seven pieces in place.

2 big triangles 1 middle triangle 2 small triangles 1 square 1 slanted piece = 7 pieces

Let us count the pieces:

  • 2 big triangles (one orange at the top, one blue on the left),
  • 1 middle-sized triangle (green, bottom right),
  • 2 small triangles (blue at the top right, orange in the middle),
  • 1 square (green, in the middle),
  • 1 slanted piece (orange, along the bottom). It has 4 sides, but it leans over, like a rectangle that has been pushed sideways. Its math name is a parallelogram (say "pa-ra-LEL-o-gram").

That is 2 + 1 + 2 + 1 + 1 = 7 pieces.

Look closely at the triangles. They are all the same shape in different sizes, like the triangles in the last lesson. Each one has one square corner and two sharp corners that are the same.

The rules of the game

There are only two rules.

  1. Use all seven pieces in every picture.
  2. Pieces must touch, but must not overlap (no piece on top of another).

You may turn the pieces any way you like. You may also flip the slanted piece over. (Flipping matters only for the slanted piece. If you flip a triangle or the square, it looks just the same!)

The secret: the small triangle is the measuring piece

Why do the pieces fit together so well? Here is the secret. Take the two small triangles. Put them together, and they can make three other pieces!

square middle triangle slanted piece
  • Two small triangles, long sides together, make the square.
  • Two small triangles, side by side, make the middle triangle.
  • Two small triangles, one moved along, make the slanted piece.

So the square, the middle triangle and the slanted piece each take up exactly as much room as 2 small triangles. They look different, but they cover the same amount of the table.

Now the big triangles. A big triangle is the same as a square and two small triangles put together:

= 3 pieces one big triangle

The square is worth 2 small triangles, and there are 2 more small triangles. So a big triangle is worth 2 + 2 = 4 small triangles.

Now we can work out the whole tangram square, using the small triangle as our measuring piece:

Piece How many Worth (in small triangles) Total
big triangle 2 4 each 8
middle triangle 1 2 2
square 1 2 2
slanted piece 1 2 2
small triangle 2 1 each 2
all pieces 7 16

The whole tangram square is worth 16 small triangles. And here is the beautiful part: every tangram picture uses all seven pieces, so every tangram picture covers the same amount of table: 16 small triangles' worth. The cat, the boat and the running person all look completely different, but they are all exactly the same size inside. Moving pieces around changes the shape of a picture, but not how much space it covers.

That idea is very important in math. When you are older, it is how people work out the space inside a shape (its area): cut it up and move the pieces into a shape you already understand.

Building pictures

Here are two simple pictures that use only some of the pieces, to get started.

a house: 3 pieces a boat: 5 pieces

A house (3 pieces). The two big triangles fit together, long sides touching, to make a big square: the walls. The middle triangle sits on top: the roof. Why does the roof fit so neatly? The long side of the middle triangle is exactly as long as the side of the big square. (You can check this with real pieces: lay them side by side.)

A boat (5 pieces). The bottom of the boat is the square with a small triangle on each side. The sails are a big triangle and the middle triangle, standing up on the boat.

How to solve a tangram picture

When you see the outline of a picture and must fill it with the seven pieces, here is a good plan.

  1. Place the big pieces first. The two big triangles are the hardest to fit. Look for big pointy parts of the outline where they could go.
  2. Look for square corners. Every triangle has one square corner. The square has four. Match them to the square corners of the outline.
  3. Use the small pieces last. The small triangles are good for filling little gaps at the end.
  4. If you get stuck, swap. Remember the secret: two small triangles can take the place of the square, the middle triangle or the slanted piece. Try swapping to see where pieces fit.

Worked examples

Example 1. Make the square from two pieces. Which two pieces make a shape exactly the same as the tangram square piece?

  1. The square is worth 2 small triangles.
  2. So try the two small triangles.
  3. Put their long sides together. Their square corners go at opposite corners of the new shape. Yes, it is a square! The two small triangles.

Example 2. How big is half the tangram? The two big triangles together cover what part of the whole tangram?

  1. Each big triangle is worth 4 small triangles. Two of them: 4 + 4 = 8.
  2. The whole tangram is worth 16.
  3. 8 is half of 16. So the two big triangles cover half of the whole tangram square. Look at the first picture: the orange and blue big triangles fill the whole top left half of the square, corner to corner. It matches!

Example 3. Do two pictures cover the same space? Ana makes a cat with all seven pieces. Ben makes a candle with all seven pieces. Ben says, "My candle is bigger, it is so tall!" Is he right?

  1. Both pictures use all seven pieces.
  2. The pieces do not overlap, so each picture covers exactly the space of the seven pieces.
  3. So both pictures cover the same amount of space: 16 small triangles' worth each.
  4. Ben's candle is taller, but it is also thinner. Taller is not the same as bigger. They cover the same space.

Try it

  1. How many pieces are in a tangram? How many of them are triangles?
  2. Which piece is the slanted one with 4 sides? What is its math name?
  3. How many small triangles is the middle triangle worth?
  4. Which is worth more: the square, or the slanted piece?
  5. How many small triangles are the two small triangles and the square worth all together?
  6. Make a big triangle using three pieces. Which three did you use?
  7. Cut out a tangram from paper (ask a grown-up to help draw the lines, using the first picture). Make the house and the boat. Then make up a picture of your own and give it a name.
  8. If you made a picture using only the two big triangles and nothing else, how much of the whole tangram would it cover?
Answers
  1. 7 pieces. 5 of them are triangles: 2 big, 1 middle and 2 small.
  2. The parallelogram. It has 4 sides, and it leans over like a pushed rectangle.
  3. 2 small triangles. Two small triangles placed side by side make the middle triangle exactly.
  4. Neither. They are worth the same: 2 small triangles each. They look different, but they cover the same space.
  5. Each small triangle is worth 1, and the square is worth 2. So 1 + 1 + 2 = 4. That is the same as one big triangle, which is why those three pieces can be put together to make a big triangle.
  6. The square and the two small triangles (see the picture in "The secret"). Another way: the middle triangle and the two small triangles also make a big triangle, because 2 + 1 + 1 = 4. Try it with real pieces!
  7. There is no single answer. Check your picture: all seven pieces used, none overlapping.
  8. Half of it. Each big triangle is worth 4, so the two together are worth 8, and 8 is half of 16.

Common mistake

Thinking a taller or wider picture is bigger. Every picture made from all seven tangram pieces covers exactly the same amount of space, however tall, wide or spread out it looks. Changing the shape does not change how much space it covers. (This is a mistake even grown-ups make, so be proud if you spot it!)

Overlapping pieces. If a piece sits on top of another, you have broken a rule, and your picture covers less space than it should.

Where it leads