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?
[Math] How to draw a regular pentagon with compass and straightedge
geometric-constructiongeometrypolygons
Related Solutions
Two circles + five lines = 7 steps:
Select an arbitrary point $P_0$ on a given circle $C_0$ and draw a circle $C_1$ with an arbitrary radius $r$, which is small enough to intersect $C_0$.
Draw a circle $C_2$ centered at the intersection point $P_1$ with the same radius $r$ to get the intersection points $P_2$ and $P_3$.
Draw a line through points $P_2,P_1$, intersecting $C_0$ at point $P_4$.
Draw a line through points $P_1,P_3$, intersecting $C_0$ at point $P_5$.
Draw a side $P_5 P_4$ of the inscribed equilateral triangle.
Draw a side $P_4 P_0$ of the inscribed equilateral triangle.
Draw a side $P_0 P_5$ of the inscribed equilateral triangle.
No, it's not possible; in fact, the regular heptagon is the regular polygon with the least number of sides that is impossible to construct with compass and straightedge alone. It is, however, possible to construct it using a neusis ruler. A related question has been asked (and answered) here.
Best Answer
I think this is the easiest way to draw a regular pentagon with just a compass and straightedge.