The goat problem is a problem in recreational mathematics known since the 18th century. It was first published in 1748 in England, in the yearly publication The Ladies Diary: or, the Woman’s Almanack.
How big must be chosen in the diagram, in order for the red area to equal one half of the area of the circle? Illustration: A goat/bull/horse is tethered at point . How long needs the line to be, to allow the animal to graze on exactly one half of the circle area?
Solution by calculating the lens area
The area of a lens with two cirles of radii and distance between centers is:
which simplifies in case of and one half of the circle area to
Solution using integration
By integrating over the right half of the lens area with
the transcendent equation
follows, with the same solution.
The goat in space
In the three-dimensional case, point lies on the surface of a unit sphere, and the problem is to find radius of the second sphere so that the volume of the intersection body equals exactly half the volume of the unit sphere.
The volume of the unit sphere reachable by the animal has the form of a three-dimensional lens with differently shaped sides and defined by the two spherical caps.
The volume of a lens with two spheres of radii and distance between the centers is:
which simplifies in case of and one half of the sphere volume to
leading to a solution of
It can be demonstrated that with increasing dimensionality, approaches the value .
The goat and the silo
In the two-dimensional case, the question about the reachable area outside the red circle may be asked. This concerns a situation where the animal is tethered to a silo.
The area consists of a half-circle (light blue) with radius and of two areas, which are bordered by the red circle and the circle involute (dark blue). Using Leibniz’s sector formula, the size of one of the dark blue areas can be calculated. The entire reachable area (light and dark blue) then equals
assuming that (otherwise, the two dark blue areas would intersect behind the silo).
- Raymond Clare Archibald: Involutes of a circle and a pasturage problem. In: American Mathematical Monthly, 1921 (28), pp. 328–329.
- Marshall Fraser: A tale of two goats. In: Mathematics Magazine, 1982 (55), pp. 221–227.
- Jean Jacquelin: Le problème de l’hyperchèvre. In: Quadrature, 2003, 49, pp. 6–12. ISSN 1142-2785