# Converting 2D density (circle) to 3D density (sphere)

Hi All,
I'm looking for help in converting 2D density (objects/area) in a circle to 3D density (objects/volume) in a sphere, the circle and sphere having the same radius and distribution of objects being uniform.

To make this problem more intuitive, here's a sample application: both crabs and fish forage on the same plankton in the water column. The crab can only access plankton as he walks on the ground (i.e., he forages in 2D space), but the fish can access the plankton in the water column (i.e., he forages in 3D space). The crab has a detection area of ∏*acuity^2 while the fish has a detection volume of (4/3)*∏*acuity^3. Given a prey density, can the number of prey within the 2D detection zone be converted to the number of prey within the 3D detection zone (or vice versa)?

I may be overthinking this, but help is much appreciated! Thanks!

haruspex
Homework Helper
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2020 Award
In your concrete example, the crab's feeding zone isn't really 2D - it's a thin sheet. Since the crab's detection range in the vertical direction can taken to extend right through th sheet you can treat it as 2D. But maybe this isn't really the scenario that you want to apply it to.

chiro