Difference between revisions of "Construct a pentagon"

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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.  
 
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.  
 +
  
 
== Prerequisites ==
 
== Prerequisites ==
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Elementary theorems about similar and isosceles triangles are also used. The ''Golden Ratio'' is discovered and defined before use.  
 
Elementary theorems about similar and isosceles triangles are also used. The ''Golden Ratio'' is discovered and defined before use.  
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== Notation ==
 
== Notation ==
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* Citations such as (P 01) refer to earlier steps in our proof.  
 
* 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.  
 
* All angles are measured in ''radians''; thus a ''right angle'' is π/2.  
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== Contents ==
 
== Contents ==
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<!-- standard sections below here -->
 
<!-- standard sections below here -->
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==Bibliography==
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* (ISBN 0764101102
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* (ISBN 0395977274
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* (ISBN 0764553240
 +
 +
 +
== Amazon's Related Products ==
 +
* (ASIN B0007PF7JW
 +
* (ASIN B00004UDSA
 +
* (ASIN B00006IB97
 +
 +
 +
== External Links ==
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* [[wp:Golden ratio|Golden Ratio]] Wikipedia Article
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* [[wp:Pentagon|Pentagon]] Wikipedia Article
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* [http://www.contracosta.cc.ca.us/math/pentagrm.htm The Pentagram & The Golden Ratio]
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[[Category:Geometry]]
 
[[Category:Geometry]]
 
[[Category:Drafting]]
 
[[Category:Drafting]]

Revision as of 19:58, 25 July 2006

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.


Prerequisites

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.


Notation

  • 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.


Contents


Bibliography

  • (ISBN 0764101102
  • (ISBN 0395977274
  • (ISBN 0764553240


Amazon's Related Products

  • (ASIN B0007PF7JW
  • (ASIN B00004UDSA
  • (ASIN B00006IB97


External Links