Line Limit - Futility Closet

Line Limit

You own a goat and a meadow. The meadow is in the shape of an equilateral triangle each side of which is 100 meters long. The goat is tied to a post at one corner of the meadow. How long should you make the tether in order to give the goat access to exactly half the meadow?

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Draw a hexagon whose sides are 100 meters long. Now draw a circle, with the same center, whose radius r is the desired length of the tether: The difference in area between the hexagon and the circle is 6 times half the area of the meadow. The area of the circle is πr². The area of an equilateral triangle with side length a is $\displaystyle \frac{a^{2}\sqrt{3}}{4}$ . The area of a hexagon with side length a is $\displaystyle \frac{6a^{2}\sqrt{3}}{4}$ . For a = 100 the area of the hexagon is $\displaystyle 15000\sqrt{3}$ . If we want a circle half that size, then $\displaystyle \pi r^{2} = 7500\sqrt{3}$ and $\displaystyle r = \sqrt{\frac{7500\sqrt{3}}{\pi }} \approx 64.3037.$ (From Ian Stewart’s Cabinet of Mathematical Curiosities, 2009.)

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September 23, 2018September 24, 2018 | Puzzles