Quandaries and Queries
Imagine that you cut you octagon into 9 pieces as in the diagram.
You get a 10 foot by 10 foot square in the middle, 4 rectangles and 4 isosceles, right angles triangles. The key is to find the length x in the diagram.
The square has area 100 square feet.
Each triangle has hypotenuse of length 10 feet and two other sides of length x feet. Thus, by Pythagoras theorem
Hence each rectangle has area 10 5 √2 square feet.
Each triangle has area 1 /2 5 √2 5 √2 square feet
Thus the area is
I get 482.8 square feet.