A (somewhat) generalisation

Geometry Level pending

Imagine a 3-D polyhedron. This polyhedron has to be a convex polyhedron and a sphere can be inscribed in this polyhedron with all of the polyhedron's faces tangent to the sphere. This sphere has radius \( r\).

By a method, let's say displacement, we get that the volume of this polyhedron is \(v\).

Therefore, what is the surface area of this shape? (Also, the material to make up this shape is not soluble in water)

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