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Do this with Gabriel's horn and you'll observe that it has an area of pi, which does not reflect the fact that its perimeter is infinite.

Yeah, I'm not sure how I concluded you could go from volume to area.

It does seem intuitive that you'd be able to, I can certainly understand how you'd think that. But no!

Nope. Edges do not work that way. Mathematically a fractal outline can have infinite length but finite area.

Consider a short portion of shoreline in Fourier space. By keeping the "DC" term constant (i.e. the integral), you have complete freedom to change all other terms. By increasing high frequency terms the length of shoreline can tend to infinity without changing the area of the island.

> the area of the edge

No, it will give you the area of the island (more precisely, the area of the sea-level cross section of the island).