How does Liu Hui calculate the area of plane figure by using cotangent principle?

The so-called principle of complementarity in and out is simply to move a plane figure from one place to another with the same area. If the graph is divided into several blocks, and the sum of the areas of each part is equal to the area of the original graph, then the sum and difference of the areas of each part before and after the graph transfer have a simple equal relationship.

The same is true for three dimensions. Give a few simple examples, as shown in the figure:

Extended data:

In Notes on Arithmetic in Nine Chapters, Liu Hui developed China's ancient thought of "rate" and the principle of "complementary entry and exit". Most of the algorithms and problems in "Nine Chapters Arithmetic" are proved by "rate", and Pythagorean theorem and some formulas for calculating area and volume are proved by the principle of "complementary access".

Liu Hui's work not only had a far-reaching impact on the development of ancient mathematics in China, but also laid a lofty historical position in the history of mathematics in the world. In view of Liu Hui's great contribution, many books call him "Newton in the history of Chinese mathematics".