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.
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.