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This classic problem from the Jiuzhang Suanshu (九章算术) is called Reed in a Pond (池中葭).

Problem

Suppose there is a square pond measuring 1 zhang wide. A reed grows at the center of the pond with 1 chi extending above the water. When the reed is drawn towards the edge of the pond, the tip just reaches the edge. Question: What is the depth of the pond and the length of the reed?

Solution

Draw a right triangle such that the reed drawn towards the edge of then pond is the hypotenuse. Let be the distance from the centre of the pond to the edge of the pond. Let be the length of the entire reed, and be the segment of the reed submerged underwater (this is also the depth of the pond). The length of the reed is 1 chi above the surface of the pond, thus it follows that .

Substitute into .

Calculation

Since ,

The depth of the pool is 12 chi, and the height of the reed is 13 chi.

As shown in Figure 5-5, let represent half the width of the pond, the depth of the pool, the length of reed, and the length of reed extending above the water. The formula given in Jiuzhang Suanshu is:

To derive the above formula, rewrite the Pythagorean theorem as

Substituting this formula into

gives

Calculation