Construct a pentagon

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This tutorial explains How to construct a regular pentagon: specifically, how to inscribe a regular pentagon within a given circle using only compass and straightedge. A fairly rigorous proof is given, generally suitable for high school math students. No trigonometry is used; all equations are of second degree or less; and the proof depends only on the most elementary geometric theorems.

For the sake of amusement, it is shown that, given the inscribed pentagon, we may also construct the pentagram (or pentacle) and two five-pointed stars.


Before attempting this construction and proof, you should have a working knowledge of:

  • Elementary Algebra
  • Euclid's Definitions, Axioms, and Postulates
  • Pythagorean Theorem (I.47)

The proof rests in part on these propositions:

  • Perpendicular Bisector (I.10, I.11)
  • Angles of any Triangle Sum to π (I.32)
  • Equal Angles Inscribed in a Circle subtend Equal Chords (III.21)

Elementary theorems about similar and isosceles triangles are also used. The Golden Ratio is discovered and defined before use.


  • Citations such as (I.1) refer to Euclid's Elements: in this case, Book I, Proposition 1.
  • Citations such as (P 01) refer to earlier steps in our proof.
  • All angles are measured in radians; thus a right angle is π/2.



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