Jiuzhang Suanshu Shaoguang 18

Problem

今有積三百步。問：為圓周幾何？ 答曰：六十步。 開圓術曰：置積步數，以十二乘之，以開方除之，即得周。

Suppose there is an area of 300 [square] bu. Question: As a circle, what would the circumference be? Answer: 60 bu. Opening the circle method: Set the area in bu, multiply by 12, then extract the square root, which gives the circumference.

Solution

The area of the circle is $A = \frac{\pi d^2}{4}$. Since the circumference of a circle with diameter $d$ is $C=\pi d$, the area can be alternatively expressed as

$$A = \frac{C^2}{4\pi}.$$

Solving for the circumference yields

$$C = \sqrt{4\pi A}.$$

The original method described in the Jiuzhang Suanshu assumes $\pi = 3$; thus,

$$C = \sqrt{12 A}.$$

Applying this formula yields

$$\begin{align} C &= \sqrt{12 \times 300} \\ [5 pt] &= \sqrt{3600} \\ [5 pt] &= 60 \end{align}$$

This explains why original text provided the answer 60 bu. A more accurate answer would be 61.4 bu.