"Sun Tzŭ's Computational Classic: Volume II" 《孫子算經卷中》 §21

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§21. Lateral surface area of a cone

This section gives a worked example of computing the lateral surface area of a cone.

The relevant unit conversions for area and length are

\begin{aligned} 1 \unit{yardland~(頃)} &= 100 \unit{acres~(畝)} \\ 1 \unit{acre~(畝)} &= 240 \unit{paces~(步)}^2. \end{aligned}

Translation

Chinese source text: Version A, Version B, Version C, Version D.
Unless noted otherwise, I follow the text from Version D, 《知不足齋叢書》本.

Source text	Target text	Notes
今有邱田、周六百三十九步、徑三百八十步。問為田幾何。	Suppose there be [a] mound field, of circumference six hundred [and] thirty-nine paces, [and] diameter three hundred [and] eighty paces. [We] ask, how much field be [this]?	Version A and Version C have the interchangeable 丘 for 邱. The "mound field" referred to appears to be a raised field in the shape of a cone. Here the "diameter" is not that of the base, but rather twice the slant height. (Indeed the circumference is less than three times the "diameter", noting that \pi \approx 3 as per Vol. I §5.)
答曰、二頃五十二畝二百二十五步。	Answer saith: two yardlands, fifty-two acres, [and] two hundred [and] twenty-five paces.
術曰、半周、得三百一十九步五分、半徑、得一百九十步。二位相乘、得六萬七百五步。	Method saith: halving [the] circumference, resulteth in three hundred [and] nineteen paces [and] five tenths, [and] halving [the] diameter, resulteth in one hundred [and] ninety paces. [The] two places multiplied with each other, result in six myriad seven hundred [and] five paces.	Version A, Version C, and Version D are missing 得 after 二位相乘.
以畝法除之、即得。	Dividing it by [the] acre divisor, [we] are done.	In modern notation, the lateral surface area of a cone with circumference C = 639 \unit{paces} and twice-slant-height D = 380 \unit{paces} is \begin{aligned} A &= \frac{C}{2} \cdot \frac{D}{2} \\[\tallspace] &= \frac{639 \unit{paces}}{2} \cdot \frac{380 \unit{paces}}{2} \div \frac{240 \unit{paces}^2}{\unit{acre}} \\[\tallspace] &= 252 \unit{acres} + 225 \unit{paces}^2 \\ &= 2 \unit{yardlands} + 52 \unit{acres} + 225 \unit{paces}^2. \end{aligned} Note that the incline of the field is an unrealistically steep \cos^{-1} \frac{C / (2 \pi)}{D / 2} = 58\degree.

Source text

Target text

Notes

今有邱田、周六百三十九步、徑三百八十步。問為田幾何。

Suppose there be [a] mound field, of circumference six hundred [and] thirty-nine paces, [and] diameter three hundred [and] eighty paces. [We] ask, how much field be [this]?

Version A and Version C have the interchangeable 丘 for 邱.
The "mound field" referred to appears to be a raised field in the shape of a cone. Here the "diameter" is not that of the base, but rather twice the slant height. (Indeed the circumference is less than three times the "diameter", noting that \pi \approx 3 as per Vol. I §5.)

答曰、二頃五十二畝二百二十五步。

Answer saith: two yardlands, fifty-two acres, [and] two hundred [and] twenty-five paces.

術曰、半周、得三百一十九步五分、半徑、得一百九十步。二位相乘、得六萬七百五步。

Method saith: halving [the] circumference, resulteth in three hundred [and] nineteen paces [and] five tenths, [and] halving [the] diameter, resulteth in one hundred [and] ninety paces. [The] two places multiplied with each other, result in six myriad seven hundred [and] five paces.

Version A, Version C, and Version D are missing 得 after 二位相乘.

以畝法除之、即得。

Dividing it by [the] acre divisor, [we] are done.

In modern notation, the lateral surface area of a cone with circumference C = 639 \unit{paces} and twice-slant-height D = 380 \unit{paces} is
\begin{aligned} A &= \frac{C}{2} \cdot \frac{D}{2} \\[\tallspace] &= \frac{639 \unit{paces}}{2} \cdot \frac{380 \unit{paces}}{2} \div \frac{240 \unit{paces}^2}{\unit{acre}} \\[\tallspace] &= 252 \unit{acres} + 225 \unit{paces}^2 \\ &= 2 \unit{yardlands} + 52 \unit{acres} + 225 \unit{paces}^2. \end{aligned}
Note that the incline of the field is an unrealistically steep
\cos^{-1} \frac{C / (2 \pi)}{D / 2} = 58\degree.

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II" §21. Lateral surface area of a cone

Translation

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《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§21. Lateral surface area of a cone