Systematic Treatise on Arithmetic, Volume Three

General Song for Measuring Land

In ancient times, measuring fields meant comparing the width and the length,
Relying entirely on ropes and rulers to pull and measure the distance.
Though every shape has its own particular method,
Only the "Square Field" method The term Fangtian (Square Fields) refers both to squares/rectangles and to the general category of area calculation. is easy to explain in detail.
If you encounter shapes that are slanted, tilted, concave, or curved,
You must "cut from the surplus to fill the deficit" The principle of yibi buqi (移彼補此), a fundamental Chinese geometric concept where shapes are rearranged into rectangles to find their area. to make them square.
Then, take the product of the dimensions as the field's area,
Divide it by 240 original: 二四 (two-four). This is shorthand for 240, as 240 square bu equal 1 mu. to make the number of mu clear.

Another Song

For a square, multiply the side by itself to find the area in square bu.
For a rectangular field, multiply the length and width together.
For right-angled triangles, isosceles triangles, and diamond shapes, multiply the dimensions and halve the result.
For a circular field, multiply half the circumference by half the diameter.
Or, square the circumference and divide by twelve This uses the approximation of pi ≈ 3..
Or, square the diameter and multiply by 0.75.
Or, multiply the circumference and diameter and divide by four.
For bowl-shaped or hill-shaped fields, use the same circular methods above.
For a ring-shaped field, add the inner and outer circumferences together,
Halve that sum, then multiply it by the width of the path.