Chapter 17 The area of ​​the second circle, curved edge and curved surface

"Nine Chapters" proposes four algorithms for the area of ​​a circle: "multiply the half-circle radii to get the product step", that is, S=1/2lr; "multiply the circumference and radius, four into one", that is, S=1/4ld; Multiply, three, four into one", that is, S=3/4d; "Multiply each other, twelve into one", that is, S=(1/12)l.Where S, l, r, and d are the area, circumference, radius, and diameter of a circle, respectively.The first two formulas are equivalent and correct in theory, but in the era of "Nine Chapters", the ratio of three diameters to one is used, and the actual calculation error is relatively large.The latter two formulas are inherently inaccurate.Liu Hui proved the first formula (see Section 11) with limit thinking, and used this formula to find out the ratio of pi=157/50, thus correcting the latter two formulas.

Gongxingtian was called Hutian in ancient times. The formula given in "Nine Chapters" is S=(1/12)(cv+v), where c is the string and v is the arrow.Liu Hui believed that this formula was inaccurate, and proposed a method of approximating Hutian with a series of triangles. The circular field is called Huantian. The algorithm proposed in "Nine Chapters" is: area S=1/2(L+L)d, where L and L are the lengths of the middle and outer circumferences, and d is the distance between the middle and outer circumferences.Liu Hui proposed a new formula S=1/2(Lr+Lr). When Liu Hui proved that the formula for the area of ​​Wantian (referring to the spherical crown-shaped field) in "Nine Chapters" was inaccurate, he proposed the formula for the area of ​​the side of the cone: S=1/2Ll, or S=1/2πdl, where L is the lower circumference, l is the length of the busbar, and d is the diameter of the lower circumference.