A regular octagon is formed from a square by making 45° cuts from each corner.

(a) Draw a diagram or construct a model. See figure below.

(b) Since the octagon is "inscribed" in the square, its area is less than that of the square.

Explain using only "Euclidean" methods why the perimeter of the octagon is less than the perimeter of the square.

(c) If the perimeter of the square is 4 show that the perimeter of the octagon is 8(√2 - 1).

Note that the perimeter of the octagon is roughly 20% less than the perimeter of the square. Reasonable?

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