Liu Hui's π algorithm: Difference between revisions

Revision as of 18:07, 1 December 2021

Revision as of 18:09, 10 January 2022

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In the diagram {{math|''d''}} = excess radius. Multiplying {{math|''d''}} by one side results in oblong {{math|ABCD}} which exceeds the boundary of the circle. If a side of the polygon is small (i.e. there is a very large number of sides), then the excess radius will be small, hence excess area will be small.

As in the diagram, when {{math|''N'' → ∞}}, {{math|''d'' → 0}}, and {{math|ABCD → 0}}.

"''Multiply the side of a polygon by its radius, and the area doubles; hence multiply half the circumference by the radius to yield the area of circle''".

When {{math|''N'' → ∞}}, half the circumference of the {{math|''N''}}-gon approaches a semicircle, thus half a circumference of a circle multiplied by its radius equals the area of the circle. Liu Hui did not explain in detail this deduction. However, it is self-evident by using Liu Hui's "in-out complement principle" which he provided elsewhere in ''The Nine Chapters on the Mathematical Art'': Cut up a geometric shape into parts, rearrange the parts to form another shape, the area of the two shapes will be identical.

Thus rearranging the six green triangles, three blue triangles and three red triangles into a rectangle with width = 3{{math|''L''}}, and height {{math|''R''}} shows that the area of the dodecagon = 3{{math|''RL''}}.