3410.32 – Squares Make An Octagon

Referring to the figure above, area of $EFGH$ is 1 square meter, so $EF = 100 \ \cm$. From this, it follows that

$$\begin{aligned}\dfrac{x}{\sqrt2} + x + \dfrac{x}{\sqrt2} &= 100 \\\dfrac{2x}{\sqrt2} + x &= 100 \\x\sqrt2 + x &= 100 \\x(1 + \sqrt2) &= 100 \\x &= \dfrac{100}{1 + \sqrt2} \\[1em]&\approx 41.42 \text{ cm}.\end{aligned}$$

The area of the octagon equals the area of EFGH minus the four shaded triangles, and we observe that those four shaded areas can be cleverly arranged to make the shaded square shown below with side $x$. The area of this square is $x^2 = 41.42^2.$ So the octagon's area is

$$100^2 - 41.42^2 \approx 8284 \text{ cm}^2$$