Mnemonic for Finding the Height and Width of a Pointed Field

Excluding the triangular tip, the shape is treated as a trapezoid.

Suppose there is a shape with a triangular top and a trapezoidal base. The upper width is 1 foot 6 inches, and the lower width is 12 feet 8 inches. The vertical height of the trapezoidal section below the triangle is 10 feet 5 inches. Question: What is the height of the triangular tip?
Answer: The height of the tip is 1 foot 5 inches.

Determining the Height and Width of a Pointed Field

This diagram illustrates a 'gui' (圭, a pointed or triangular shape) placed atop a 'ti' (梯, a trapezoid). It shows how to calculate the height of the apex based on the proportions of the widths below. This type of geometry was essential for surveyors calculating the area of irregular plots of land.

The method states: Take the vertical height 10 feet 5 inches and multiply it by the upper width 1 foot 6 inches to obtain 16 feet 8 inches as the dividend (shi: the total value to be divided). Separately, subtract the upper width 1 foot 6 inches from the lower width 12 feet 8 inches; the remainder of 11 feet 2 inches acts as the divisor (fa: the measure used to divide). Dividing the dividend by the divisor yields the height of the triangle's tip as 1 foot 5 inches, which fits the question.

Mnemonic for Finding the Lower Width of a Pointed Field

A mathematical diagram illustrating a 'gui' (triangular/trapezoidal) shape calculation. It contains internal labels in traditional Chinese script and a horizontal line dividing the tip (圭尖) from the base (下廣).

Previous page translation for continuity:

Obtaining the width difference 6 paces and subtracting it from the lower width 22 paces leaves 16 paces, which is the middle width that fits the question.

General Source of Computational Methods, Volume 7

Page 4

Suppose there is a trapezoidal field original: 梯田 (titian). A field shaped like a ladder, which is the traditional Chinese term for a trapezoid. with an area of 1,500 paces, a northern width The "northern" width refers to the top horizontal side of the trapezoid in a standard orientation. of 40 paces, and a central length of 50 paces. What is the southern width?

The method states: Take the area 1,500 paces and double it to get 3,000 paces as the dividend (shi: the total value to be divided). Use the length 50 paces as the divisor (fa: the measure used to divide). Dividing results in 60 paces. From this, subtract the northern width 40 paces; the remainder is the southern width of 20 paces, which fits the question.

Suppose there is an original slanted field with a southern width of 4 paces, a northern width of 10 paces, and a length of 12 paces. Now, one wishes to extend it into the form of a right-angled triangle gougu: the traditional Chinese mathematical term for a right triangle, where "gou" is the base and "gu" is the height...