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Square Barn

My grandson's high school trigonometry teacher posed him a fun geometry problem: A goat is tethered by a 50 ft. rope to one corner of a 10 foot square barn. What is the area the goat can graze? Dougy and I worked on it for about three hours and found the answer. Can you figure it out? What approach would you use? What happens at the corners? It helps to have a grasp of high school math and geometry. Yes, you can use your scientific calculator.
The length of rope and dimensions of barn are immaterial, the process is the same for any numbers.

As I looked through the internet to check our solution, I found several variations on the theme, and I decided to come up with a general solution to be used with any regular polygon. Some examples are given on the conclusions page. For this answer start with Step 1