Tethered Goats
Geometry & Trigonometry
A goat is tethered by a rope to the wall of a shed. What area of grass can it reach, and where should the rope be tied to maximise it?
Introduction
This is a classic sector-geometry problem, once used in an IB SL exam. A goat is tethered by a rope to a point on the outside wall of a rectangular shed. As it moves, keeping the rope taut, it sweeps out a region made of circular sectors, one at each corner it can reach around. No specific shed dimensions or rope length are given here; choose your own and build the scenario from scratch.
Guiding Questions
- A goat is tethered to the corner of a shed. Sketch and calculate the area it can graze.
- How does the grazing area change as the rope gets longer than the shed walls?
- Solve the inverse problem: what rope length gives the goat exactly half the field?
- Move the anchor point or change the shed shape. Which setup is hardest, and why?