In this case I am imagining a light source which shoots rays in all directions and forms a sphere and I would like to know how to calculate what percentage of the sphere's surface area is intercepted by a square solar cell with a surface area of 9 square inches.
Assuming I position the cell 5 inches away, (perpendicular to the centre of the cell), I can't just do $\frac{9}{100 \pi}$, dividing SA of cell by SA of sphere.This image shows a light source with rays pointing out of it in all directions.
The percentage of surface area of the light sphere that is blocked by the square cell is given by:
$ \arctan \left(\frac{S^2}{2d^2 \sqrt{4 d^2 + 2S^2} } \right) \frac{100}{\pi}$
where S is the side length of the square cell and d is the distance from the centre of the cell to the centre of the sphere. Ref Wikipedia article.
Putting the numbers in we get:
$ \arctan \left(\frac{3^2}{2(5)^2 \sqrt{4 (5)^2 + 2 (3)^2} } \right) \frac{100}{\pi} \approx 2.631$ %.
As a sanity check imagine the cell is 1.5 inches away from the source. Six such cells arranged as a cube could then completely enclose the sphere and absorb 100% of the light. One such cell at this distance would then absorb (100/6)% $\approx$ 16.66%.
Entering d = 1.5 inches in the formula given above, does indeed gives the correct percentage (16.666).