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Problem

《數書九章·田域類2》

問沙田一段,有三斜,其小斜一十三里,中斜一十四里,大斜一十五里。 里法三百步,欲知為田幾何?

Shushu Jiuzhang

There is a field of sand with three sides. The short side is 13 li, the medium side is 14 li, and the long side is 15 li. The li rule is 300 bu. How large is the field?

Qin Jiushao's triangle2.png

Solution

In the Shushu Jiuzhang, Qin Jiushao gave the formula:

where is the area of a triangle, is the short side, is the middle side, and is the long side. 

The area Qin Jiushao gave is 315 qing. This is calculated by knowing that 1 li is 300 bu, 1 mu is 240 sq.bu, and 1 qing is 100 mu.

Derivation

Triangle with notations 3.svg.png

Let be the base, be the height, and be the area of the triangle, then

From the above diagram, determine two equations for :

Equation 1

Equation 2

Equate both equations and solve for :

Substitute this result into Equation 1 and take the positive square root:

The area of the triangle is therefore

or

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