Spheres in Cubes

In the above figure, the spheres are touching each other. I'm trying to derive a formula that tells me how much space of the cube the spheres occupy when the radius of the center sphere is varied. Unfortunately, I'm having trouble doing this...

To begin, each side of the cube is length s. Therefore, the volume of the cube is s^3. The volume of any sphere is 4/3 pi r^3.

Now let's define the outer semispheres as having radius 1/2, and the inner sphere having a radius of k/2. Now to relate s and k, the cube has to be divided down a diagonal, which looks like the following:



One of the sides of this rectangle is sqrt(2) s, and the other side is s. The diagonal of this rectangle is therefore sqrt(3) s. Since the diagonal of this rectangle is the length of twice the inner radius plus twice the radius of the outer radius:

sqrt(3) s = 2 (k/2+ 1/2) = k+1

s = (k+1) / sqrt(3)

From here, the total volume of the two spheres divided by the total volume of the cube should give the formula. (Each of the 8 outer semispheres adds up to one sphere, and the inner sphere is one sphere.)

[4/3 pi (1/2)^3 + 4/3 pi (k/2)^3] / [(k+1) / sqrt(3)]^3

pi/6 (1+k^3) / [(k+1) / sqrt(3)]^3

[sqrt(3) pi /2] (1+k^3) / (k+1)^3

Unfortunately, this does not give me the formula I was looking for. When k = 0, the function should spit out p/6. Instead, it spits out sqrt(3) pi/2. Halp.