Orbology

The unit radius sphere of surface area $S$ is partitioned by 2 parallel planes into 3 solid parts of equal surface area.
Let the surface area of one of these three parts be $T$ .
The sphere of surface area = $S$ - $T$ has diameter $D$ .

If $D = a + b \sqrt{c}$ , where $a, b, c$ are integers and $c$ is square-free, then find $a + b + c$ .