Tangram questions, Triangles, and squares...?

Question

Does anyone know if there is a square made of 6 tangram pieces, also is it possible to make a triangle out of 4,5,6,7 tangram pieces...I only have 2 large triangles, 1 medium triangle, 2 small triangles, 1 small square, and a small parallelogram. Think you can figure this out?! GOOD LUCK!

Answer 1

I don't think it's possible to make a square out of 6 of the pieces.

Let's consider the areas of the pieces: If we define the length of the side of the small square as one, then the areas are as follows:

small square: 1
small triangle: 1/2
parallelogram: 1
medium triangle: 1 (the length of it's leg is sqrt(2)
large triangle: 2
The combined area of all the pieces is 8.

Now consider which piece you are going to leave out.
If we leave out a large triangle, then the total area is 6. That means the sides of the square would have to be sqrt(6). We only have side lengths in rational numbers and multiples of sqrt(2), so we can't make sqrt(6)

Leaving out the square, the medium triangle, or the parallelogram gives us a total area of 7 and we have the same problem.

Leaving out a small triangle gives us a total area of 15/2. The length of a side would have to be sqrt(30)/2. We can't make this either.

I think this proves it is impossible.

If you know how to make a square with all seven pieces (like this: http://images.google.com/images?q=tbn:DH55RoYJ6EkYeM:http://g3400.nep.chubu.ac.jp/onsenkids/kagakutoy/tangram/tangram.gif
then you can see how to make a triangle with 5 pieces... it's just half of the square.

You can also modify the 7-piece square into a seven-piece triangle by separating it into two triangles and setting them together so that they are leg to leg and the other legs form the new hypotenuse.

You can make a triangle with 4 pieces using a large triangle, two small triangles, and the parallelogram. Start with the large triangle. Along one leg, place the short side of the parallelogram so that the parallelogram forms a right angle with the acute angle of the large triangle. Then put one small triangle between the long side of the parallelogram and the large triangle and put the other small triangle with its leg adjacent to the other short side of the parallelogram.

I'm not sure about triangles made with 6 pieces. I think it probably can't be done for similar reasons that the square can't be done. I'm pretty sure it would have to be a right triangle, so you could probably do a similar area-based analysis.

Answer 2

Tangram Triangle

Answer 3

How do you make a square with the 5 smallest tangram pieces?