Dividing a Triangle into Four-Sided Sections

The method states: Set each side at 14. Multiply by 7 and divide by 8 This is a traditional Chinese approximation for the height of an equilateral triangle. Using the ratio 7:8, 14 paces results in a height of 12.25, though the text rounds this to 12 for simplicity in the following steps. to obtain a middle diameter height of 12 paces.

Alternatively, multiply each side of 14 by the diameter of 12 to get 168 square paces. Halve this to obtain the area of 84, which serves as the shi: dividend (the number to be divided). Divide this by three sections, which yields 28 paces square paces for each. Then, halve each side [of the original triangle] to get 7 paces. Divide [the section area of 28] by this 7, and double the result to obtain the middle length of 8 paces. This matches the inquiry. This problem describes dividing an equilateral triangle into three equal quadrilaterals by connecting the center point to the midpoints of the sides.

Cutting a Triangular Piece from a Rectangular Field

Suppose there is a rectangular field 15 paces long and 12 paces wide. Now, based along the width, a gui: a pointed isosceles triangle shape with an area of 45 square paces is to be cut out. The question asks: What is the length of this triangular cut?

The answer states: The length of the triangular cut is 7.5 paces original: "7 paces and 5 fen." One pace contains 10 fen..

The method states: Take the area to be cut and double it to get 90 as the shi: dividend. Use the width of 12 paces as the fa: divisor. Dividing [the dividend by the divisor] yields the result.
The remaining methods for cutting trapezoids and other shapes are all illustrated in the diagrams to the left.

Rectangular field (直田), Triangular shape (圭), Width (濶), Middle diameter or height (中徑), Area (積), Dividend (實), Divisor (法), Divide/Reduce (歸), Halve (折半)