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Rex and Ralph: Polygonal House
Category:  Math 
Submitted By:  javaguru 
Fun:  (2.4) 
Difficulty:  (3.22) 
Two mathematicians, Rex and Ralph, have an ongoing competition to stump each other. Ralph has just finished building a custom house and invites Rex to dinner.
He tells Rex, "The lot my house sits on is a regular polygon. My house is a matching regular polygon sitting on a circular foundation in the middle of the yard. The closest part of the foundation to the edge of my property is exactly the same as the diameter of the foundation. The house is two stories, built around a central circular atrium with a diameter that is exactly one tenth the longest measurement of my property."
"Sounds like quite the house!" remarked Rex.
"Yes, I've been building it for almost two years. When I was excavating the foundation I found a section of an old irrigation pipe. The pipe exactly bisected the area of the house, with each end of the pipe at an edge of the house. After the house was completed while I was preparing to put up a fence around the edge of the property, I discovered that the length of the pipe I found was exactly the length of one side of the property. I used the pipe as a gate across the side of the property with the driveway and built the fence around the other sides. I built a total of 400 feet of fence."
"So what's the square footage of my house?" Ralph asked.
"Are you counting the space occupied by the walls in the square footage?" asked Rex.
"Yes of course," Ralph replied. "It also includes the stairways, hallways, and basically all of the space inside of the exterior walls."
Rex smiles and says, "Nice try, Ralph! Assuming the information you gave me is correct, it'll take just a minute or two to calculate it."
How did Rex know the square footage, and what was his answer?
Hint:
A regular polygon has the length of all sides and arc of all interior angles equal.
The first thing you need to determine is the number of sides for Ralph's house. The rest solves in a fairly straightforward manner once this is determined.
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