Difference between revisions of "Pentagon"

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 $540^{\circ}$.

Construction

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 $36^{\circ}$.\\