By: Tao Steven Zheng (郑涛)

【Problem】

Jiuzhang Suanshu (Gougu 5)

Suppose there is a tree 2 zhang tall whose circumference is 3 chi. A kudzu vine grows from the bottom of the tree and coils around the tree 7 times, such that the top reaches the top of the tree. Question: How long is the kudzu vine?

Note: 1 zhang is 10 chi

【Solution】

The problem requires the Pythagorean theorem to compute the length of a helical vine. As shown in the figure above, let be the height of the tree, be the circumference of the tree multiplied by the number of coils of the tree, and be the length of the kudzu vine. It is known that the height of the tree is 2 zhang, or 20 chi. The base of the right triangle is 7 times the circumference of the tree. Since the circumference of the tree is 3 chi, the base is 21 chi. When unwound, the length of the vine is the hypotenuse of the right triangle. 

Hence,

The vine measures 29 chi, or 2 zhang 9 chi, in length.