A cross-section through a cube can be an equilateral triangle, a square, and a regular hexagon, as seen in my drawing below.
It can also be a number of less regular shapes. But is it possible for it to be a regular pentagon?
Solution in comments.
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Friday 19 September 2025 at 21:33
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Edited by Richard Walker, Saturday 20 September 2025 at 13:50
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It's possible for a section through a cube to be a pentagon but not a regular one. The reason is this.
When you take a section through a polyhedron the result is a polygon with a side corresponding to each polyhedral face you have sliced through. For a pentagon we must cut through five faces, and these must include two pairs of opposite faces.
But if a plane cuts through two other planes that are parallel to one another the intersection is two lines which are parallel. So, the polygonal section must have two pairs of parallel sides, which makes it impossible for it to be a regular pentagon, none of whose sides are parallel.
Notice that similar considerations apply to a hexagonal section: it must have three pairs of parallel sides, but this obviously allows it to be a regular hexagon.