Systematic Treatise on Arithmetic, Vol. 6

Song for Cutting Area from a Triangular Field

If dividing a field into three segments, first calculate the cut from the pointed tip; the lower two segments are then calculated using the method for trapezoids.

To find the tip cut from a triangular field,
Double the cut area and multiply by the original length.
Divide this by the original width to find the shi: dividend (the base number for calculation);
Extract the square root to find the appropriate length of the cut.
Then, multiply this cut length by the original width,
And use the original length as the divisor.
The result of this division is the width of the cut section;
The method is clear and simple—do not doubt it.

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Suppose there is a triangular field 75 paces long with a northern width of 30 paces. Now, a section with an area of 405 square paces is to be cut from the pointed tip. The question asks: What are the length and width of this cut?
Answer: The length is 45 paces; the width is 18 paces.

The method states: Set the cut area of 405 paces and double it to get 810. Multiply this by the original length of 75 to get 60,750 paces. Divide this by the original width of 30 to get 2,025, which serves as the shi: dividend for extracting the square root.

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Cutting the Tip of the Triangle

Original length: 75 paces

Original width: 30 paces

Original length: 75 paces Original width: 30 paces Area of the cut section: 405 square paces

The term "Gui field" (圭田) refers to a field shaped like a ceremonial stone tablet—essentially a triangle. The math here relies on the principle of similar triangles, where the ratio of the areas is proportional to the square of the ratio of the lengths.