Systematic Treatise on Arithmetic, Vol. 8

If you take a circle's diameter of 18 feet and use 1 foot 2 inches original: "一尺二寸"; in this context, it acts as a constant ratio of 1.2 to divide it, you obtain a square's side literally "square diameter" of 15 feet.

If you take a square's side of 15 feet and multiply it by 1 foot 2 inches, you obtain a circle's diameter of 18 feet.

Song for Finding the Base, Height, and Hypotenuse from their Differences
The "difference" refers to the variance between sides.
This refers specifically to the difference between the height and hypotenuse, or the base and hypotenuse.

To find the height from the height-difference: square the base.
Subtract the square of the height-difference from the squared base.
The remainder of the base after subtraction is the dividend 實 (Shi): the value to be divided.
Twice the height-difference serves as the divisor 法 (Fa): the number to divide by.
Divide the dividend by the divisor to find the value of the height.
Finding the base from the base-difference is achieved in the same way.

○ To find the hypotenuse from the hypotenuse-difference: square the base.
Divide the result by the hypotenuse-difference to find the "true state" an intermediate value in the calculation.
Add the hypotenuse-difference to this and halve the result;
The length of the hypotenuse is then successfully calculated.

Suppose there is a base 勾 (Gou) with a width of 27 steps. It is only stated that the hypotenuse 弦 (Xian) is 9 steps longer than the height 股 (Gu). What are the lengths of the height and hypotenuse?

Answer: The height is 36 steps; the hypotenuse is 45 steps.

Using the second mnemonic:
1. Square the base: 27 × 27 = 729.
2. Divide by the difference: 729 ÷ 9 = 81.
3. Add the difference: 81 + 9 = 90.
4. Halve it: 90 ÷ 2 = 45 (this is the hypotenuse).
5. Subtract the difference to find the height: 45 - 9 = 36.