On a recent1 episode of Wrong, But Useful, Dave mentioned something interesting2: if you take three regular shapes that meet neatly at a point – for example, three hexagons, or a square and two octagons – and make a cuboid whose edges are in the same ratio as the number of sides on each shape (e.g., 6 by 6 by 6 or 4 by 8 by 8), the resulting cuboid has a volume that’s (numerically) the same as its surface area (here, 216 or 256).
Let’s prove it!
The interior angle of a regular $n$-gon is $\pi – \frac{2n}{\pi}$, or $180 – \frac{360}{n}$ if you insist on silly angle measures. In fact, it’s…
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The Maths Echo Chamber 