How long a rope lets a goat eat exactly half the grass in a triangular field?
A goat is tethered to one corner of a half-acre field shaped like an equilateral triangle, and can graze right to the end of its rope. How long must the rope be — to the nearest inch — so the goat can eat exactly half the grass in the field?
Reveal the answer
About 1,731 inches (48 yards and 3 inches). Six equilateral triangles the size of the field would tile together into a regular hexagon, so the goat's grazing arc — a 60° circular sector, since the field's corner angle is 60° — covers exactly one-sixth of a full circle. Setting that sixth-circle equal to a quarter-acre (half of the half-acre field) and solving for the radius gives the rope length.
— Henry Dudeney, Amusements in Mathematics — Thomas Nelson & Sons, 1917, Puzzle No. 196
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