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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?

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    Euclid did too, *much* earlier. Do you want Gauss's construction, or any? And have you tried looking around the web, for example, [Wikipedia](https://en.wikipedia.org/wiki/Pentagon)? – 2017-01-18
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    @pjs36 ANY construction, but Gauss's would be great. – 2017-01-18
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    http://math.stackexchange.com/questions/1281221/construct-a-regular-pentagon-in-only-11-steps-using-ruler-and-compass/1281273#1281273 – 2017-01-18
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    The 19 year old Gauss showed how to construct [a regular 17-sided polygon](https://en.wikipedia.org/wiki/Heptadecagon), and later characterized [which regular polygons are constructible](https://en.wikipedia.org/wiki/Constructible_polygon). – 2017-01-18

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

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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: