3410.32 – Squares Make An Octagon

Two congruent squares of side length 1 meter are placed so that their overlap is a regular octagon. What is the area of the octagon?

Solution

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

\begin{aligned} \frac{x}{\sqrt2} + x + \frac{x}{\sqrt2} &=& 100 \\ \frac{2x}{\sqrt2} + x &=& 100 \\ x\sqrt2 + x &=& 100 \\ x(1 + \sqrt2) &=& 100 \\ x &=& \frac{100}{1 + \sqrt2} \\ &\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

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