Notices
Results 1 to 12 of 12

Thread: A Formula for Computing Diagonals for a Regular Polygon

  1. #1 A Formula for Computing Diagonals for a Regular Polygon 
    Guest
    Let us consider any regular n-gon. It must follow that all of the angles are congruent. Similarly, all of the angles must have an equal number of diagonals before any are constructed.

    Let us consider an arbitrary edge: it will have n - 3 diagonals that can be constructed, since the sides that form its vertex and its own cannot be drawn. It follows that the next vertex will also have n - 3 diagonals capable of construction (because it could be regarded as the side of the previous vertex). If one continues to construct the diagonals, the remaining vertices will have n - 4, n - 5,... 1 vertices remaining (because the vertices cannot be regarded as the sides that formed the previous vertex considering all vertices have a set of sides that are unique).

    Ergo, the series is as follows:





    We can use the above formula to find the diagonals of any regular polygon.


    Reply With Quote  
     

  2.  
     

  3. #2 Re: A Formula for Computing Diagonals for a Regular Polygon 
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by Ellatha
    Let us consider any regular n-gon. It must follow that all of the angles are congruent. Similarly, all of the angles must have an equal number of diagonals before any are constructed.

    Let us consider an arbitrary edge: it will have n - 3 diagonals that can be constructed, since the sides that form its vertex and its own cannot be drawn. It follows that the next vertex will also have n - 3 diagonals capable of construction (because it could be regarded as the side of the previous vertex). If one continues to construct the diagonals, the remaining vertices will have n - 4, n - 5,... 1 vertices remaining (because the vertices cannot be regarded as the sides that formed the previous vertex considering all vertices have a set of sides that are unique).

    Ergo, the series is as follows:





    We can use the above formula to find the diagonals of any regular polygon.
    this has nothing to do with the regularity of the n-gon. If by "diagonals" you mean line segments joining vertices that lie completely within the figure, all that is needed is for the n-gon to be convex, which regular n-gons are.

    More simply the counting problem can be posed in terms if the number of edges of a complete graph with n vertices. The number of edges of a complete graph with n vertices is clearly

    =

    Now the number of edges of a n-gon is just so the number of "diagonals" is simply

    =

    Your expression is not wrong, but merely clumsy since



    =

    =


    Reply With Quote  
     

  4. #3  
    Forum Ph.D. Heinsbergrelatz's Avatar
    Join Date
    Aug 2009
    Location
    Singapore
    Posts
    994
    I had a similar question to this in my maths textbook last year, including the topic mathematical induction.
    ------------------




    "Mathematicians stand on each other's shoulders."- Carl Friedrich Gauss


    -------------------
    Reply With Quote  
     

  5. #4  
    Guest
    I wrote that when I was about fifteen or so and just found the paper so I obviously did not understand how to simplify summations.
    Reply With Quote  
     

  6. #5  
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by Ellatha
    I wrote that when I was about fifteen or so and just found the paper so I obviously did not understand how to simplify summations.
    Are we to the conclude that have have learned little in the interim ? The post was made yesterday. Did you simply copy old work mindlessly ? Why ?

    That particular sum is one of the very first that are learned when students first learn about sums and the "sigma" notation. There is a famous (though likely false) story about Gauss finding the answer very quickly in response to a school assignment asking for the sum of the first (some large number) integers.

    He did it simply as follows

    x = 1 + 2 + 3 + ... +N
    x = N + (N-1) + (N-2) + ... + 1
    ___________________________

    2x = (N+1) + (N+1) + ........+(N+1)

    2x = N(N+1)

    x= N(N+1)/2
    Reply With Quote  
     

  7. #6  
    Guest
    Quote Originally Posted by DrRocket
    Are we to the conclude that have have learned little in the interim ? The post was made yesterday. Did you simply copy old work mindlessly ? Why ?
    I'm not quite sure I follow what you're saying: I have about two notebooks dedicated to any research I've done in mathematics alone over the past few years (starting at about when I was fourteen). For a while I believed that most of the ideas were my own, only to find that they were discovered by mathematicians several centuries earlier. I believe there may still be some that are perhaps new. I post them on this forum mostly for discussion.

    The first paper I ever wrote was the following, rudimentary one:

    "If one takes any natural number composed of a single digit, regardless of how large the number is, and multiplies it by nine, the result will always be composed nines with the digit to the farthest left and farthest right making up the product of that number and nine."

    For example, consider the number 88888888888888888888888888888888 multiplied by nine. The digit to the farthest left will be a seven, and the digit to the farthest right will be a two. The remaining will be nine.

    I provided a rather humorous proof related to the fact that any single digit number multiplied by nine will have the sum of the units and ten digits equal to nine.

    "That particular sum is one of the very first that are learned when students first learn about sums and the "sigma" notation. There is a famous (though likely false) story about Gauss finding the answer very quickly in response to a school assignment asking for the sum of the first (some large number) integers."

    I am familiar with that story of Gauss: it is a rather amusing one.

    An even more interesting story is one of a student at the University of Michigan:

    After his professor had explained to the class a theorem that had alluded him and his colleagues for years, the student wrote the theorem down before the class ended.

    A week or so later he left hundreds of sticky notes on the professors desk. The student had proved the theorem. He would also go on to be the Unabomber.
    Reply With Quote  
     

  8. #7  
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by Ellatha
    I'm not quite sure I follow what you're saying: I have about two notebooks dedicated to any research I've done in mathematics alone over the past few years (starting at about when I was fourteen). For a while I believed that most of the ideas were my own, only to find that they were discovered by mathematicians several centuries earlier. I believe there may still be some that are perhaps new. I post them on this forum mostly for discussion.
    Lots of people have simple mathematical ideas that have occurred to them. Very few keep such meticulous records of trivia. Clutter reduction is a good thing.

    http://www.kitchensource.com/trash/
    Reply With Quote  
     

  9. #8  
    Guest
    Quote Originally Posted by DrRocket
    http://www.kitchensource.com/trash/
    Buy me one.
    Reply With Quote  
     

  10. #9  
    Forum Ph.D. Heinsbergrelatz's Avatar
    Join Date
    Aug 2009
    Location
    Singapore
    Posts
    994
    Quote Originally Posted by Ellatha
    I wrote that when I was about fifteen or so and just found the paper so I obviously did not understand how to simplify summations.
    When I was 15, so about 1 and a half years ago, I certainly wasn't the best at dealing with sums but definitely i could have done the sums DrRocket has posed, and Way more.


    http://www.kitchensource.com/trash/

    for a split second i was thinking, is DrRocket for real? a literal website for trashcans? and i was right.
    ------------------




    "Mathematicians stand on each other's shoulders."- Carl Friedrich Gauss


    -------------------
    Reply With Quote  
     

  11. #10  
    Guest
    Want a cookie?
    Reply With Quote  
     

  12. #11  
    Forum Ph.D.
    Join Date
    Jul 2008
    Location
    New York State
    Posts
    846
    When I was in high school (stuyvesant high, NYC) many many years ago I wrote an article for the school math journal describing the same problem. To the best of my recollection I came up with n(n-3)/2 in a rather straightforward manner, i.e. each vertex can be connected to n-3 other vertices. Divide by 2, since each diagonal gets counted twice.
    Reply With Quote  
     

  13. #12  
    Forum Ph.D. Heinsbergrelatz's Avatar
    Join Date
    Aug 2009
    Location
    Singapore
    Posts
    994
    Quote Originally Posted by Ellatha
    Want a cookie?
    sure
    ------------------




    "Mathematicians stand on each other's shoulders."- Carl Friedrich Gauss


    -------------------
    Reply With Quote  
     

Bookmarks
Bookmarks
Posting Permissions
  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •