@qed.maths: Post 93 | Push a stack of coins sideways until it leans at 45°, or even make it snake: every coin is still the same coin, so every slice keeps its area πr², and by Cavalieri the volume stays πr²h. But the leaning stack needs more paint. Cut it square to its axis and the section is an ellipse with semi-axes r and r cos α, while the stack is now L = h / cos α long. Its side is that ellipse's perimeter times L: 15.28 against 12.57 for the upright stack (r = 1, h = 2), 21.6% more skin for exactly the same volume. Cavalieri protects the volume, never the surface. #maths #geometry #cavalieri #cylinder #ellipse

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Tuesday 29 September 2026 11:01:46 GMT