The Right-Angled Triangular Field

Gougu field (gōugǔ tián): A field in the shape of a right-angled triangle. "Gou" refers to the shorter leg (base) and "Gu" refers to the longer leg (height).

Suppose a right-angled triangular field has a height of 60 steps and a base width of 32 steps. What is the area?

Height: 60 steps

Base width: 32 steps

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Answer: 960 square steps.

Method: Set the height of 60 steps and multiply it by the base width of 32 steps to get 1,920 steps. Halve this result to obtain 960 steps The area of a triangle is calculated as (base × height) / 2. This matches the query.

The Rectangle via the Sum of Sides

Suppose a rectangular field has a combined width and length of 92 steps, and the diagonal distance between the two corners is 68 steps. What is the area?

Combined width and length: 92 steps

Diagonal distance: 68 steps

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

Method: Take the diagonal of 68 steps and multiply it by itself to obtain 4,624 steps.
Separately:

Take the combined sum of 92 steps and multiply it by itself original: "自乘" (zì chéng) — to square a number to obtain 8,464 steps.
Subtract the smaller from the larger:

Subtract the smaller result from the larger to get a remainder of 3,840 steps. Halve this to obtain the correct area. This algebraic shortcut works because (Width + Length)² minus (Diagonal)² equals twice the Area (2WL).

**Technical