It's been 45 years since I took high school geometry, and I still have not been able to find the solution for creating a regular pentagon with compass and straight-edge. A 72 degree angle with old-fashioned Euclidean methods. Does anybody here know how to do it?
Hello Fraggle Rocker! 45 years, eh? It has been 2 years since I had geometry and I probably know much less than you Please Register or Log in to view the hidden image! I am an algebra person... Because of my lack of geometry power, I had to be creative. Maybe this method has been used before? Note that a regular pentagon has an angle of 72 degrees. Assuming you do not know anything about radians, I will just leave it in degrees but I would rather be in radians Please Register or Log in to view the hidden image! 1)Draw a circle with your compass with a radius of r. The plan is to draw the regular pentagon in the circle, with each vertex on the circle. 2) Using some basic trig. (I remember covering some in geomtry), you can derive the formula L = 2*r*tan(angle/2). The L is the length of the sides of the pentagon. 3) Once you have the length, measure that out on the compass. Then, placing both ends on the circle, make arcs to mark the vertex of the circle. Where the arcs intersect the circle is where the vertex is. I hope this helps. Please Register or Log in to view the hidden image! It might not be the solution you were looking for, but like I said,... I never payed attention in geometry. James Sibley
But that doesn't answer Fraggle Rocker's question which was about constructing a pentagon with straightedge and compasses. I did a "google" search for 'construction "regular pentagon" ' and found this site: http://216.239.37.100/search?q=cach...contruction "regular pentagon"&hl=en&ie=UTF-8
That's what I thought but I was waiting for confirmation. It seems to beg the question of knowing the tangent of 36 or 72 degrees. Under the Euclidean rules, the only way to know any of the trigonometric functions of an angle is to draw it and measure them with a compass and straight edge. It doesn't display well, at least not on my computer using IE. A lot of mathematical symbols are missing so there's no way I can follow the reasoning. They seem to imply that the key to the problem is that the cosine of 72 degrees is a rational number, but it sure doesn't look like one in the online trig tables. I understand radians. 360 degrees = pi radians. Radians are useful in calculus but I've never seen them used in trig before. In trig you can divide a full circle by 3 or 5 and then divide the resulting angle by 2 (and maybe by 3?) as many times as you want. 360 degrees with all of its factors is an easier number to work with than pi. I've done a Google search as well, and I still can't find a legible proof for inscribing a regular pentagon.
360 degrees = 2pi randian, not pi randians Please Register or Log in to view the hidden image! I like radians a lot because they make more sense than degrees. As far as making a pentagon with a compass and a straight edge, I found a link. http://mathforum.org/library/drmath/view/54752.html James Sibley
Of course. That's why they're called "radians" instead of "diametrians." I guess it's been a long time since I've used this stuff. Oh wait, I'm an "elder." I'm supposed to be telling the kids, "Be sure and study your geometry and trigonometry. You'll find those skills to be invaluable in real life." Thanks for both of the websites. The Canadian university was the easiest to read and understand intuitively, although I doubt that I'll remember it without actually going through the construction a few times until I can do it from memory. Dr. Math supplied the symbol that was missing in the earlier website. The formula involves the square root of five, which of course is the hypoteneuse of a 1x2 right triangle. It was a shorter proof and will probably be easier to memorize, but it's not so intuitively sensible. Well I have to run now. I have a lifetime of projects that are stalled, waiting for regular pentagons, that I can now get back to. Please Register or Log in to view the hidden image!
Well, one math teacher was honest with everyone: 'Most people do not run around doing logarithms all the time.' Please Register or Log in to view the hidden image! What kind of probjects? Try my method of making a pentagon. I would like to see my creativity in action. I do not have a compass, so I cannot try it myself :-/
