# ag.algebraic geometry – How do I find or derive circumspherical radii, surface area, and volume for these 5 non-traditional concave polyhedra?

As much of a mathematics enthusiast I am, I’m stuck on a huge roadblock in further developing my code for the paper that shows the value in different geometries at the nanoscale for nanomedicine: There are five shapes that I am having trouble deriving the circumspherical radii, and thus, the equations of surface area and volume, to solve my equations:

Medial Rhombic Triacontahedron
Dodecadodecahedron
Medial Triambic Icosahedron
Ditrigonal Dodecadodecahedron
Excavated Dodecahedron

I’ve scoured the internet to find some approximations but the only one I could find was based off of one of those solids so far and it’s not a modifiable equation as I cannot denote where the side length comes into play (they are just constants based off of a unit edge length): http://dmccooey.com/polyhedra/MedialRhombicTriacontahedron.html

I was wondering if anyone knew how to derivate the shape’s equations or where I can find the correct resource? I’d derivate it myself but it’s a bit above my scope of knowledge. I need a physical solid to cut and manipulate to try to find basic ratios at the very least.

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