### III. The Three-Width Field

Suppose a field has a straight length of 86 steps. What is its area?
Answer: 2,494 square steps.

Method: Add the North and South widths together and halve the sum to get 40 steps. Add the middle width to this for a total of 58 steps. Multiply this by the length to get 4,988 steps. Halve this result to obtain the total area original: "積" (jī) — the accumulation or product of dimensions, used here for area.

This matches the query.

Alternative Method: Double the middle width and add the North and South widths together for a total of 116 steps. Divide this by four original: "歸" (guī) — a term for division, specifically referring to abacus-based division tables to get 29 steps. Multiply this by the length to obtain the same result.

North width: 30 steps
Width: Middle width: 18 steps
South width: 50 steps

Note: A Three-Width Field is essentially the combination of two trapezoidal fields trapezoidal field (tì tián): literally "ladder field," referring to a four-sided shape with two parallel sides of different lengths. The distances between the three measured widths must be equal for the Three-Width Method to be applied calculate (suàn): to compute using mathematical principles or an abacus. If the upper section is long and the lower section is short, or the upper is short and the lower is long, the Three-Width Method cannot be used in its entirety. Instead, one must calculate the two trapezoids individually and add them together to ensure there are no errors.

Also note: Drum-shaped fields, Waisted Drum fields, Arrow-clamp fields, and Arrow-fletching fields likewise require the distances between the three widths to be equal to use this method. If the distances are unequal, simply calculate based on three widths or as two trapezoids and combine them.