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?

Euclid did too, much earlier. Do you want Gauss's construction, or any? And have you tried looking around the web, for example, Wikipedia? — pjs36, Jan 18 '17 at 03:51
http://math.stackexchange.com/questions/1281221/construct-a-regular-pentagon-in-only-11-steps-using-ruler-and-compass/1281273#1281273 — Joffan, Jan 18 '17 at 05:02
The 19 year old Gauss showed how to construct a regular 17-sided polygon, and later characterized which regular polygons are constructible. — hardmath, Jan 18 '17 at 14:03

score 9 · Answer 1 · answered Jan 18 '17 at 11:17

9

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

answered Jan 18 '17 at 11:17

Seyed

8,933

this is not a regular pentagon – NutronStar45 Feb 22 '24 at 15:12

score 2 · Answer 2 · answered Aug 19 '20 at 04:36

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.

score 1 · Answer 3 · edited Feb 24 '17 at 21:29

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

Draw a circle. Let the center be $O$.
Define a direction as "left" and draw a line from the center going "left" until you hit the circle. This segment is $OA$.
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 $OB$.
Find the midpoint of $OA$, calling it $M$.
Draw $BM$.
Find the angle bisector of $BMO$ and draw until you hit $OB$. Call this intersection $I$.
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:

How to draw a regular pentagon with compass and straightedge

3 Answers3