Keeping with my longing running theme of talking about the pound coin and shapes of constant diameter here is a nice little result known as Barbier’s theorem.

Simply, it says that for any curve of constant diameter (width; \(w\)) the perimeter of that shape is \(\pi\times w\). A nice result. The proof follows from Crofton’s formula.

Interestingly though the result doesn’t hold for higher dimensional shapes – e.g. the surface area for solids of constant width depends on more than the width alone…

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