# Construct a pentagon

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.

## Contents

## 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

**Problems****Construction****Proof****Miscellany**

## Bibliography

- (ISBN 0764101102
- (ISBN 0395977274
- (ISBN 0764553240

## Amazon's Related Products

- (ASIN B0007PF7JW
- (ASIN B00004UDSA
- (ASIN B00006IB97

## External Links

- Golden Ratio Wikipedia Article
- Pentagon Wikipedia Article
- The Pentagram & The Golden Ratio