The side length of the equilateral triangle inside the bounded region is 1/3, so the area of this equilateral triangle is \(\displaystyle \frac{\sqrt{3}}{4}\cdot\left(\frac{1}{3}\right)^2 = \frac{\sqrt{3}}{36}\). Each of the four smaller regions outside this equilateral triangle is bounded by a copy of the Koch curve that has been scaled by 1/3, so each of them has area A/9. Therefore \[A = \frac{\sqrt{3}}{36} + 4\cdot\frac{A}{9} \Rightarrow \frac{5}{9}A = \frac{\sqrt{3}}{36} \Rightarrow A = \frac{\sqrt{3}}{36}\cdot \frac{9}{5}= \frac{\sqrt{3}}{20}.\]
More generally, if the length of the initial chord is \(L\), then the area bounded by the Koch curve is \(\displaystyle \frac{\sqrt{3}}{20}L^2\).
