text: how do you construct a regular pentagon and a regular decagon? can you construct a regular n-gon?   i assume that you are wanting straightedge and compass construction.  to make a decagon, let o be the center of a circle, with boa one diameter and cod a perpendicular diameter (if you can't construct this, you shouldn't be doing this construction.) bisect oa at the point e. now draw a circle centered at e of radius ed. let this circle intersect the line bo at the point f. then of is the length of the side of a decagon. mark off arcs ofthis radius around the original circle.  for a pentagon, you can either just skip every other vertex from the decagon (easy!) or it turns out that the side length is the same as the distance fd above.  the problem of which n-gons can be constructed by ruler and compass was solved by gauss when he was 17 years old. (!) for polygons with a prime number of sides, the only ones that are constuctible are those which are one more than a power of 2 (the fermat primes): so far the only ones known are 3,5,17,257,65537.  then, the only constructible n-gons are ones where n is a power of 2 times a (possibly empty) product of fermat primes with no repeats. so, 15=3*5 is possible, but 9=3*3 is not. nor are 7,11,13,14,25, etc. however, all of 4,8,16,48,51,761, 204 are.  hope this helps!