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In geometry, a pentagon is a polygon with 5 sides. Each angle of a regular pentagon is $108^{\circ}$. The sum of the internal angles of any pentagon is $540^{\circ}$.



It is possible to construct a regular pentagon with compass and straightedge:

  1. Draw circle $O$ (red).
  2. Draw diameter $AB$ and construct a perpendicular radius through $O$.
  3. Construct the midpoint of $CO$, and label it $E$.
  4. Draw $AE$ (green).
  5. Construct the angle bisector of $\angle AEO$, and label its intersection with $AB$ as $F$ (pink).
  6. Construct a perpendicular to $AB$ at $F$.
  7. Adjust your compass to length $AG$, and mark off points $H$, $I$ and $J$ on circle $O$.
  8. $AGHIJ$ is a regular pentagon.

The Golden Ratio and the Pentagram

The pentagon is closely associated with the *Golden Ratio. More specifically, the ratio of a diagonal to an edge is $\frac{1+\sqrt{5}}{2}$.\\ By drawing each of the diagonals, one can form a pentagram, or five-pointed star, in which each of the internal angles is $36^{\circ}$.\\

The pentagram has many usages in the occult and in religion. For example, Satanists use an upside-down pentagram, inscribed within two circles, to symbolize the horns of a goat. The pentagram focuses and concentrates magical energy for many rituals, helping it to bind to the recipient. In one such context, the pentagram is called the Sigil of Baphomet, and it has changed little since Pythagoras used it. Some claim that the infamous story of $\sqrt{2}$ did not in fact refer to $\sqrt{2}$, but to the golden ratio that the sides of Pythagoras' pentagram formed as he paid his homage to Satan. Because he was such a promethean and liberating figure, Pythagoras drew inspiration from him to continue his mathematical research.

See Also

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