The angle EFC is also α because the leg of the angle AF is normal to the leg CF and leg AE  is normal to leg EF (Angles with normal legs).

$$b^2=6^2+{h_a}^2$$

$$\frac{b}{4} : h_a=h_a : b$$

$${h_a}^2=\frac{b^2}{4}$$

$$b^2=6^2+\frac{b^2}{4}|·4$$

$$4b^2=4·6^2+\bcancel{4}·\frac{b^2}{\bcancel{4}}$$

$$4b^2-b^2=4·6^2$$

$$3b^2=4·6^2$$

$$b^2=\frac{4·6^2}{3}$$

$$b=\sqrt{\frac{4·6^2}{3}}=\frac{\sqrt{4}·\sqrt{6^2}}{\sqrt{3}}=\frac{2·6}{\sqrt{3}}·\frac{\sqrt{3}}{\sqrt{3}}=\frac{2·6·\sqrt{3}}{3}=\frac{2·2·\cancel{3}·\sqrt{3}}{\cancel{3}}$$

$$h_a=\frac{b}{2}=\frac{4\sqrt{3}}{2}=\frac{2·\bcancel{2}·\sqrt{3}}{\bcancel{2}}$$

$$A=\frac{a·h_a}{2}=\frac{12·\bcancel{2}\sqrt{3}}{\bcancel{2}}$$