The area of the region between the tractrix and the base line can be determined in a remarkably simple way. Image all the tangent segments of the tractrix translated without rotation so the points of tangency are brought to a common point. Each tangent segment has length L so the other endpoint will lie on a circle of radius L. As the child walks from its initial position to another point on the base line, the tangent segments sweep out a sector of a circle of radius L, as suggested by the following diagram.

Tangent segments to a tractrix sweep out a region whose area is that of a circular sector

As the child continues to walk along the base line, the tangent segments become more and more horizontal, and the sector becomes closer and closer to one-quarter of a circular disk.

So it is reasonable to conclude that the area of the entire region under the tractrix is exactly equal to one-quarter the area of the disk. This conclusion, which seems intuitively obvious, can be justified by the methods of calculus.