Roundest regular solid
You might reasonably suppose that having more sides would result in a rounder figure, so a figure with 20 sides should be rounder than a figure with 12 sides. But it’s the other way around.
Here’s one way to see the statement above. In a dodecahedron, three pentagons come together at each vertex. The interior angles of a pentagon have 108°. If you were to take the three pentagons, cut a slit long one slide, and push them flat on to a plane, there would be a gap of
In an icosahedron you have five triangles coming together at a point. So if you were to push the five triangles at a vertex down to a point, there would be a gap (angle defect) of
For completeness, the angle defect of a cube is 90°. For an octahedron it’s 240° and for a tetrahedron it’s 270°.
Angle defect is the standard way to measure how round a polyhedron is. It focuses on how faces fit together at a vertex. But we could instead focus on how faces fit together at an edge. The angle between two adjacent faces is the dihedral angle.
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