(related to Problem: The Tethered Goat)

This problem is quite simple if properly attacked. Let us suppose the triangle $ABC$ to represent our half-acre field, and the shaded portion to be the quarter-acre over which the goat will graze when tethered to the corner $C.$ Now, as six equal equilateral triangles placed together will form a regular hexagon, as shown, it is evident that the shaded pasture is just one-sixth of the complete area of a circle. Therefore all we require is the radius ($CD$) of a circle containing six quarter-acres or $1\frac 12$ acres, which is equal to $9,408,960$ square inches. As we only want our answer "to the nearest inch," it is sufficiently exact for our purpose if we assume that as $1$ is to $3.1416,$ so is the diameter of a circle to its circumference. If therefore, we divide the last number I gave by $3.1416,$ and extract the square root, we find that $1,731$ inches, or $48$ yards $3$ inches, is the required length of the tether "to the nearest inch."


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Project Gutenberg

  1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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