Tangram

Area

The tangram allows us to build images with all sorts of shapes, but they all have something in common: their area. Since for each image all (and only!) the tangram pieces are used, in a non-overlapping manner, the image's area is always the same.

The decomposition of figures is one of the most simple and intuitive methods to compare areas of polygons. Hence, if two flat images can be built with the same pieces, without overlapping any of the pieces, they will have the same area. With this in mind, there are, sometimes, some pseudo-paradoxes associated with tangram. Consider the images below:

The images are at the same scale and both can be built using the tangram. However, with a less careful observation, the second image appears to have a bigger area than the first one. Where did those feet come from, in the second person?...

There are various geometric pseudo-paradoxes that arise because of an inattentive eye, since images have a big suggestive power... On another page from the website – Mathematics without words– you can not only get a better sense of the power of images on the explanation of the results, but also learn to take a closer look to some geometric "paradoxes".

Note that the general and clear notion of area is a subtle concept that will not be addressed here. However, for polygonal figures, it is possible to prove the following: two images have the same area if and only if it is possible to decompose any of those images into a finite number of polygons and redistribute them in order to obtain the other image. It is important to understand that this equivalence is not obvious and that, for example, it would not be correct to apply it in the case of the volume of polyhedrons in 3D space.