Overturned Moon

Namely, the Arc and Sagitta original: 弧矢

This is a flat semi-circle, attached to the sagitta

Suppose there is an "overturned moon" field fuyue: literally "overturned moon," a poetic term for a semi-circle where the chord is 56 steps long and the sagitta shi: literally "arrow," the technical term for the height of an arc is 28 steps deep. What is the area in square steps?

Answer: 1,176 square steps.

The method states: Take the chord of 56 steps and add the sagitta of 28 steps to get 84 steps. Halve this to get 42 steps. Then multiply this by the sagitta of 28 steps to obtain the area. Calculation: [(56 + 28) ÷ 2] × 28 = 1,176. This specific formula was commonly used to estimate the area of a semi-circle.

Another method: Multiply the chord and the sagitta together 56 × 28 = 1,568. Separately, multiply the sagitta by itself 28 × 28 = 784. Add these two results together 1,568 + 784 = 2,352 and halve the sum to also obtain the area 2,352 ÷ 2 = 1,176.

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Arc and Sagitta

This explains the method for a semi-circle

This differs from the previous "Flat Ox" method; it relates to the void

Suppose there is an "arc and sagitta" field hushi: a circular segment, resembling a bow (arc) and arrow (sagitta) where the chord is 40 steps long and the sagitta is 8 steps wide. What is the area in square steps?

Answer: 192 square steps.

The method states: Take the chord of 40 steps and the sagitta of 8 steps and add them together to get 48 steps. Halve this to get 24 steps, then multiply by the sagitta of 8 steps to obtain the area as requested. Calculation: [(40 + 8) ÷ 2] × 8 = 192. This formula, (Chord + Sagitta) ÷ 2 × Sagitta, was the standard traditional Chinese approximation for the area of a circular segment.