Create extra triangles to calculate?

A side of an icosogon extends $\dfrac {360^\circ}{20} = 18^\circ$ . Let the center of the circumcircle of the icosogon be $Z$ and its radius be $r$ . Since there are 8 sides from $E$ to $M$ , therefore, $\angle EZM = 18^\circ \times 8 = 144^\circ$ , 7 sides from $M$ and $T$ , $\angle MZT = 126^\circ$ and $\angle EZT = 90^\circ$ . Then the area of $\triangle EMT$ is:

$\begin{aligned} [EMT] & = [EZM] + [MZT] + [EZT] \\ & = \frac 12r^2 \left(\sin 144^\circ + \sin 126^\circ + \sin 90^\circ \right) & \small \color{#3D99F6} \text{Note that }r = \frac {0.5}{\sin 9^\circ} \\ & = \frac 12 \left( \frac {0.5}{\sin 9^\circ}\right)^2 \left(\sin 144^\circ + \sin 126^\circ + \sin 90^\circ \right) \\ & \approx \boxed{12.2} \text{ cm}^2 \end{aligned}$