imc

There are four marked points on the diagram; let us examine the top two points and call them $A$ and $B$. Similarly, let the bottom two dots be $C$ and $D$, as shown: This is a cross-section of the sphere seen from the side. We know that $AO=BO=CO=DO=2$, and by Pythagorean Theorem, length of $\overline{AB}=2\sqrt{2}.$ Each of the four congruent semicircles has the length $AB$ as a diameter (since $\overline{AB}$ is congruent to $\overline{BC},\overline{CD},$ and $\overline{DA}$), so its radius is $\frac{2\sqrt{2}}{2}=\sqrt{2}.$ Each one's arc length is thus $\pi \cdot \sqrt{2}=\sqrt{2}\pi .$ We have $4$ of these, so the total length is $4\sqrt{2}\pi =\sqrt{32}\pi$, so thus our answer is $\boxed{32}$ Note: TLDR: The radius of $2$ gives us a line segment connecting diagonal vertices of the semi-circles with a measure of $4$, giving us through $45^{\circ }-45^{\circ }-90^{\circ }$ relations and Pythagorean theorem a diameter for each semi-circle of $2\sqrt{2}$, which we can use to bash out the circumference of a full circle, multiply by $2$, and move inside and under the root to get $32$.