I am lucky to have a new course with only seven students. However, four of these students got the answer to this problem rather quickly. The remaining three had the weekend to think about it, but got nowhere. Insert Unifix Cubes. I've had these for a million years.
Students who needed to taped them together like this:
And saw that there were 7 cubes in all that were radioactive...the first cube itself, and the other 6 that radiated from the 6 faces of said cube. So this was 7 cubic units so far.
Then, they cut paper and taped it around the cubes. Students saw that this created 3 4-quarter cylinders, so this added 3 cylinders with radius and height both equal to 1, so π cubic units. These quarter-cylinders came from the edges of the inside cube that radiated radioactivity as a radius of 1 between any two cube faces.
They then realized that they were missing the triangular pieces, shown above. This was from the corner of the middle cube, and it created eight 1/8 spheres, so 1 full sphere with radius 1. Or, 4/3π.
So the total answer is 7+3π+4/3π, or approximately 20.6 cubic units of radiation.
Some visually talented students drew it like this:
The 7 cubes
The cylinders and spheres built in
Some final results
Students were very proud of themselves either way!