Systematic Treatise on Arithmetic, Vol. 3

[Sum] Rectangular Field: Sum of the Width and Diagonal

Suppose there is a rectangular field with a length of 60 steps, and the combined sum of its width and diagonal is 100 steps. What is the area?

The sum of the height and hypotenuse is the same original: "股弦和同" (gǔ xián hé tóng). In this context, the problem treats the rectangle's width and diagonal like the base and hypotenuse of a triangle. Width: 60 steps Diagonal: steps? Length: steps?

Answer: 1,920 square steps. Halve the result, just as in the calculation for a right-angled triangular area

Method: Set the sum of the width and diagonal, 100 steps, and multiply it by itself To square the number: 100 × 100 to obtain 10,000 steps. Separately, take the width Actually refers to the known side, the length of 60 steps and square it to obtain 3,600 steps. Subtract the smaller value from the larger to get a remainder of 6,400 steps. Halve this to get 3,200 steps, which serves as the dividend. Divide this by the sum of the width and diagonal (100 steps) to find that the width is 32 steps. Finally, multiply this by the length of 60 steps to obtain the area of 1,920 square steps. This matches the query.

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[Difference] Rectangular Field: Difference Between Length and Width

Suppose the diagonal distance between two corners of a rectangular field is 68 steps, and the only other information is that the length exceeds the width by 28 steps. What is the area?

Diagonal: 68 steps Length exceeds width by 28 steps

Answer: 1,920 square steps. Halve the result, just as in the calculation for a right-angled triangular area

Method: Set the diagonal of 68 steps and multiply it by itself to obtain 4,624 steps. Separately, take the 28 steps by which the length exceeds the width and multiply that by itself to obtain 784 steps. Subtract the smaller value from the larger to get a remainder of 3,840 steps. Halve this result to obtain the area, which matches the query. This geometric shortcut relies on the fact that (Diagonal)² minus (Length - Width)² equals twice the area of the rectangle.

Rectangular field (zhítián): A field with four right angles.
Length (zòng): The vertical or longer dimension of the field.
Width (guǎng): The horizontal or shorter dimension of the field.
Diagonal (xié): The straight line connecting opposite corners; equivalent to the hypotenuse in triangle geometry.
Area (jī): Literally "accumulated [steps]," referring to the total surface measure.
Gou-Gu (gōugǔ): The Chinese name for the legs of a right triangle; by extension, the Pythagorean theorem.
Hypotenuse (xián): The side opposite the right angle.
Sum (hé): The result of addition.
Difference (chā): The result of subtraction.
Square (zìchéng): To multiply a number by itself.
Halve (zhébàn): To divide by two.