《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§13. Area of a circle

This section gives a worked example of computing the area of a circle.

The relevant unit conversion between area and length is

1 \unit{acre~(畝)} = 240 \unit{paces~(步)}^2.

See Vol. I §1 (Units of length).

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
今有圓田、周三百步、徑一百步。問得田幾何。 Suppose there be [a] circular field, of circumference three hundred paces, [and] diameter one hundred paces. [We] ask, how much field resulteth?
答曰、三十一畝奇六十步。 Answer saith: thirty-one acres remainder sixty paces.
  • 奇： remainder

    奇、音羈、 Cantonese: kei1, Mandarin: jī

    To convert area from square paces to acres, mu (畝), one divides by the conversion factor 240 \unit{paces}^2 / \unit{acre}, called the "acre divisor" below. This is why the text refers to a "remainder".

  • In modern notation, the area is 31 \unit{acres} + 60 \unit{paces}^2.
術曰、先置周三百步、半之、得一百五十步。又置徑一百步、半之、得五十步。相乘、得七千五百步。 Method saith: first put [down the] circumference three hundred paces; halving it, resulteth in one hundred [and] fifty paces. And put [down the] diameter one hundred paces; halving it, resulteth in fifty paces. [These] multiplied with each other, result in seven thousand five hundred paces.
以畝法二百四十步除之、即得。 Dividing it by [the] acre divisor two hundred [and] forty paces, [we] are done.
又術、周自相乘、得九萬步。以一十二除之、得七千五百步。 Also [a] method: [the] circumference multiplied with itself, resulteth in nine myriad paces. Dividing it by twelve, resulteth in seven thousand five hundred paces.
  • Version B has 又術曰 for 又術, both here and below.
以畝法除之、得畝數。 Dividing it by [the] acre divisor, resulteth in [the] number of acres.
又術、徑自乘、得一萬。以三乘之、得三萬步、四除之、得七千五百步。 Also [a] method: [the] diameter multiplied [with] itself, resulteth in one myriad. Multiplying it by three, resulteth in three myriad paces, [and] dividing it [by] four, resulteth in seven thousand five hundred paces.
以畝法除之、得畝數。 Dividing it by [the] acre divisor, resulteth in [the] number of acres.
  • In modern notation, a circle with circumference C and diameter d has area
    \frac{C}{2} \cdot \frac{d}{2} \approx \frac{C^2}{12} \approx \frac{3d^2}{4}.
    Here \pi \approx 3, see Vol. I §5.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §13". <https://yawnoc.github.io/sun-tzu/ii/13> Accessed yyyy-mm-dd.