A square of side length 4 cm has its corners cut away in such a manner so as to form a regular octagon. What is the length of the side of the octagon?
1.64 cm
Let x be the leg cut off at each corner. The cut edge (octagon side) is \(x\sqrt{2}\) and the remaining middle part of the square's side is \(4 - 2x\), which must also equal the octagon side: \(4 - 2x = x\sqrt{2} \Rightarrow x = \frac{4}{2+\sqrt{2}}\). The octagon side \(s = x\sqrt{2} = 4(\sqrt{2}-1)\). Using \(\sqrt{2} = 1.41\), \(s = 4(0.41) = 1.64\ \text{cm}\).
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