How to Make a Regular Pentagon from a Paper Knot
Cut a strip of paper of constant width, tie a simple overhand knot, tighten it and carefully flatten it. Trim off the ends and you are left with a pentagon. Surprisingly, it is a regular pentagon: all five sides and all five angles are equal, and simple geometry can prove it.
Why does a paper knot make a regular pentagon?
The secret is the width of the strip. Wherever two strips of equal width cross, they form a rhombus, a shape with four equal sides, because every altitude of that shape equals the width of the paper. A flattened knot contains several such crossings, and comparing them one by one shows that all five sides of the pentagon are equal. A second step, using the trapezoids hidden inside the pentagon, shows that its angles are equal too.
Key facts
- The strip must have a constant width, with parallel top and bottom edges.
- Two crossing strips of equal width always form a rhombus.
Where can I read the full story?
This is a short version of “A Knotty Problem” from OYLA 12+, Vol. 12, Issue 10 (October 2026). The full article in OYLA 12+ walks through both proofs step by step, with diagrams.
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