I have a barn that is a regular hexagon. Each side of the barn is 100 feet long. I tether my goat to a point A, a vertices of the hexagon, with a 150-ft rope. Find the area of the region in which my goat can graze.

Guest May 28, 2020

#1**0 **

Each interior vertex angle of a regular hexagon is (n - 2)·180^{o}/n = (6 - 2)·180^{o}/6 = 120^{o}.

I'll break up the area into three sections.

There is one major section, going 150' along one side in a circular arc to 150' along the adjacent side.

Since the interior angle is 120^{o}, the exterior angle will be 240^{o}.

The area of this section will be: (240^{o}/360^{o})·pi·radius^{2} = (2/3)·pi·150^{2} = 15,000 pi

Then, on each end, around the corner of the barn, the goat can go in a circular arc with radius = 50'.

This angle will be 60^{o}, or one-sixth or a circle.

The area of each section will be (1/6)·pi·50^{2} = 416 2/3 pi

Total area: 15,000 pi + 416 2/3 pi + 416 2/3 pi = 833 1/3 pi square feet.

geno3141 May 28, 2020