《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§17. Volume of a trapezoidal prism (1)

This section gives a worked example of computing the volume of a trapezoidal prism.

The relevant unit conversion for length is

1 \unit{rod~(丈)} = 10 \unit{rules~(尺)}.

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
今有隄、下廣五丈、上廣三丈、高二丈、長六十尺、欲以一千尺作一方。問計幾何。 Suppose there be [an] embankment, of lower breadth five rods, upper breadth three rods, height two rods, [and] length sixty rules, [and we] wish to use one thousand rules doing one block. [We] ask, how many amounteth [this] to?
  • 一千尺： one thousand rules

    In modern language these would be cubic rules.

答曰、四十八方。 Answer saith: forty-eight blocks.
法曰、置隄上廣三丈、下廣五丈、并之、得八丈、半之、得四丈。 Method saith: put [down the] embankment's upper breadth three rods, [and] lower breadth five rods; combining them, resulteth in eight rods, [and] halving it, resulteth in four rods.
以高二丈乘之、得八百尺。以長六十尺乘之、得四萬八千。 Multiplying it by [the] height two rods, resulteth in eight hundred rules. Multiplying it by [the] length sixty rules, resulteth in four myriad [and] eight thousand.
以一千尺除之、即得。 Dividing it by one thousand rules, [we] are done.
  • In modern notation, the volume of a trapezoidal prism of upper breadth A = 3 \unit{rods}, lower breadth B = 5 \unit{rods}, height H = 2 \unit{rods}, and length L = 60 \unit{rules} is
    \begin{aligned} V &= \frac{A + B}{2} \cdot H L \\[\tallspace] &= \frac{30 \unit{rules} + 50 \unit{rules}}{2} \times 20 \unit{rules} \times 60 \unit{rules} \div \frac{1000 \unit{rules}^3}{\unit{block}} \\[\tallspace] &= 48 \unit{blocks}. \end{aligned}

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §17". <https://yawnoc.github.io/sun-tzu/ii/17> Accessed yyyy-mm-dd.