We shall see that not all regular polygons are constructible with straightedge and compass, although most are (see the article The heptadecagon: fact and fiction). Here are a few constructions for inscribing a polygon in a circle with center O and radius R.
The hexagon
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The construction of a hexagon is familiar to every schoolchild who has ever set out to draw rosettes. Simply mark off the circle's radius to construct a regular hexagon. The circle with center A and radius R intersects the original circle at B and C. Triangle OAB is equilateral. Reflecting A, B and C gives the hexagon ABDEFC, which divides into six equilateral triangles.
The equilateral triangle
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