Pentagon Problem
How will you inscribe a pentagon in a circle
Labels: mathemagic
posted by Rajesh Lal at
9:46 PM
Monday, July 13, 2009
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13 Comments:
Very Carefully.
Actually, a regular pentagon should fit perfectly into a circle...
Actually every regular shape (triangle, square, hexagon, octagon, decadon, etc) will fit perfectly inside a circle. I believe the question is how to draw a pentagon inside a circle perfectly, which i'm not sure right now.
This post has been removed by the author.
All 3 of you are correct. (the three above me obviously)
take the perimiter of the circle devide by 5, then start at one point and put points at the number u got when u devided, then connect the dots
sry about spelling, didnt rly check
http://www.cut-the-knot.org/pythagoras/pentagon.shtml
Start with a circle of radius 1 (Figure 1).
Construct perpendicular diameters AB and CD. Bisect the radius OB at M. With center M and radius MC, draw an arc intersecting radius OA at N. I claim that CN is the side of the regular pentagon inscribed in the circle.
OK... let's be honest. You have to assume that instruments are available on hand (pencil, ruler, ect.).. Furthermore, let's agree that no matter how complicated or brilliant one's method for accomplishing this task, it is true that only a computer could create a perfect pentagon inside of a circle (and there would even be some error associated with that). Having said all of this, I will assume that I have a basic instrument available, a protractor. With this tool, the task becomes elementary.
Obviously, a pentagon's sides are of identical length. It's angles are also exactly the same. It then stands to reason that that points of intersection of the two objects must be equally spaced along the perimeter of the circle. Since degrees are, in this context, a way to partition a circle's perimeter, we can easily use our protractor to accomplish that task.
A circle has 360 degrees... divide 360 by 5. This yields 72 degrees. Now construct a circle of equal diameter as the said protractor. Tic 72 degree marks on the circle. Now use the flat end of the protractor to connect consecutive tic marks. I think this is as easy as it gets. This tedious explanation is overkill.
Jesse
jesse right idea, but pentagon each angle is 108 degrees. If all the angles is a triangle = 180. and a square is = 360. then all the angles added up in a pentagon = 540 degrees. divde that by 5 and you get each angle equaling 108. then draw your pentagon
Anon above and Jesse both have similar ideas using a protractor, but are doing it different ways. Both methods will draw a perfect pentagon though.
A bit late, but it is obvious that you are not allowed a protractor. The usual rules are compass, and straightedge only (and pencil).
This is a famous problem I believe first solved by Ptolemy. Also done by Euclid and Gauss.
Ooops. I've just noticed that Ragknot had provided a solution.
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