compass and straightedge construction of regular pentagon

One can construct a regular (http://planetmath.org/RegularPolygon) pentagon with sides of a given length $s$ using compass and straightedge as follows:

1. Draw a line segment of length $s$. Label its endpoints $P$ and $Q$.

2. Extend the line segment past $Q$.

3. Erect the perpendicular to $\overrightarrow{PQ}$ at $Q$.

4. Using the line drawn in the previous step, mark off a line segment of length $2s$ such that one of its endpoints is $Q$. Label the other endpoint as $R$.

5. Connect $P$ and $R$.

6. Extend the line segment $\overline{PR}$ past $P$.

7. On the extension, mark off another line segment of length $s$ such that one of its endpoints is $P$. Label the other endpoint as $S$.

8. Construct the midpoint of the line segment $\overline{RS}$. Label it as $M$. (Below, $\overline{PS}$ is drawn in red, and $\overline{MR}$ is drawn in green.)

   Note that the length of the line segment $\overline{MR}$ is ${\displaystyle \frac{1+\sqrt{5}}{2}}s$, which is the length of each diagonal of a regular pentagon with sides of length $s$.

9. Separately from the drawing from the previous steps, draw a line segment of length $s$.

10. Adjust the compass to the length of $\overline{MR}$ and draw an arc from each endpoint of the line segment from the previous step so that the arcs intersect.

11. Adjust the compass to the length of $\overline{PS}$ and draw arcs from each of the three points to determine the other two points of the regular pentagon.

12. Draw the regular pentagon.

The law of cosines can be used to justify this construction. Note that, in the picture below, the lengths of the line segments drawn in red are $s$ and the lengths of the line segments drawn in green are ${\displaystyle \frac{1+\sqrt{5}}{2}}s$. The color of these line segments is based off of how the pentagon above was constructed.

If you are interested in seeing the rules for compass and straightedge constructions, click on the provided.