How to draw a regular pentagon with compass and straightedge

Question

I remember reading that Gauss managed to construct a regular pentagon with just a compass and straightedge, but I don't remember the particulars of how he did this. Could someone help me out and give me instructions on how to do this?

Answer 1

I'm not sure if this one is Gauss', but here's the one I use:

1. Draw a circle. Let the center be $O$.

2. Define a direction as "left" and draw a line from the center going "left" until you hit the circle. This segment is $OA$.

3. Draw another line segment, this time going "up" (this is perfectly legal - you should know how to construct a perpendicular line to a segment). This segment is $OB$.

4. Find the midpoint of $OA$, calling it $M$.

5. Draw $BM$.

6. Find the angle bisector of $BMO$ and draw until you hit $OB$. Call this intersection $I$.

7. Draw a perpendicular line to $OB$ going "left" until you hit the circle at a point $C$. $BC$ is now one line segment of the pentagon and the rest is relatively simple (just draw circle centered around $C$ passing through $B$ to get third vertex, etc.)

Something like this:

Answer 2

I think this is the easiest way to draw a regular pentagon with just a compass and straightedge.

Answer 3

I only know two constructions. The first I learned in high school and involves straightedge and compass:

The other I learned in college as I studied compass-only (Mohr-Mascheroni) constructions. Ironically the compass-only construction is one of the simplest constructions Ive seen, only requiring 10 circles, the line-segments at the end are cosmetic.