Rolling wheel!

The trick is to figure out that M does not stay in the same place.
So, the wheel moves forward by $\dfrac{2\pi r}{4} = \dfrac{\pi r}{2}$ as it is clear from the figure.

Applying Pythagoras' theorem:-
$\begin{aligned} MM' & = \sqrt{r^2 + {\left(\dfrac{\pi r}{2} + r\right)}^2}\\ \\ & = \sqrt{r^2 + {\left(\dfrac{\pi r + 2r}{2}\right)}^2}\\ \\ & = \sqrt{r^2 + r^2{\left(\dfrac{\pi + 2}{2}\right)}^2}\\ \\ & = \sqrt{r^2\left(1 + {\left(\dfrac{\pi + 2}{2}\right)}^2\right)}\\ \\ & = r\sqrt{{\left(\dfrac{\pi + 2}{2}\right)}^2 + 1}\\ \\ \text{Putting the values}:-\\ \\ MM' & = \dfrac{3}{2}\sqrt{7.609}\\ \\ & = 1.5 × 2.758\\ \\ & \approx \color{#3D99F6}{\boxed{4.13}} \end{aligned}$