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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.

 May 28, 2020
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Each interior vertex angle of a regular hexagon is (n - 2)·180o/n  =  (6 - 2)·180o​/6  =  120o.

 

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 120o, the exterior angle will be 240o.

The area of this section will be:  (240o/360o)·pi·radius2  =  (2/3)·pi·1502  =  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 60o, or one-sixth or a circle.

The area of each section will be (1/6)·pi·502  =  416 2/3 pi

 

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

 May 28, 2020

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