1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Recreational problem involving circles, cords, and areas

  1. Jan 31, 2014 #1
    Hello all,

    I've been trying to work out this problem I came across yesterday when a professor mentioned it in a math education course. It states, "Take a circle and put two dots on the circle then connect them with a cord. How many sections of area does the cord split the circle into?" Of course the answer is 2.

    Now do it with 3 dots. The answer is 4.
    Now do it with 4 dots. The answer is 8.
    Now do it with 5 dots. The answer is 16.
    How many if there are 6 dots?

    One might think, naively as I did, that the formula for the number of sections is 2n-1, so 6 dots should have 32 sections. But it actually has 31. That was the point of the lesson.

    We never got the formula so I've been trying to figure one out.

    I decided to look at the number of dots and how many cords they form, and I was able to find a recursive definition and conjecture a formula. I proved it using induction.

    I will write the information out like this: # of dots / #of cords;

    1 / 0
    2 / 1
    3 / 3
    4 / 6
    5 / 10
    6 / 15
    7 / 21
    8 / 28

    The recursive series can be understood as s(n)=s(n-1) + (n-1), and the explicit formula is:

    s(n)=(n-1)+(n-2)+...+ 3+2+1.

    Now I have a nice relationship between dots and cords. Next I will show you the table of dots to cords to sections, which I worked out manually up to n=7.

    1 / 0 / 1
    2 / 1 / 2
    3 / 3 / 4
    4 / 6/ 8
    5 / 10 / 16
    6 / 15 / 31
    7 / 21 / 58
    8 / 28 / ???

    So this is where I am stuck. I am trying to find a relationship with the number of sections and the number or cords or dots. I do not see any obvious recursive relationship, and certainly do not see any obvious formula.

    Perhaps a different set of eyes/synapses on the problem would shed more light. Is there a more systematic approach I can use, rather than jus trying to "see" it?
  2. jcsd
  3. Jan 31, 2014 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    In this context, 'cord' should be 'chord'.
  4. Jan 31, 2014 #3
    lol I'm a musician so I thought to myself, "It can't be chord!"
  5. Jan 31, 2014 #4
    If you put the six points equally spaced (and draw it carefully enough) then you get 30 not 31 (the small one in the center disappears). So, I'm not sure you can make an algorithm to count them.

    edit: how can I get an attachment to show as a little thumbnail, so people don't have to open it to see it?

    Attached Files:

  6. Jan 31, 2014 #5
    Wow. Mind blown! Entire class came up with 31, but difficult to draw perfect circles and perfectly straight lines.

    Ok, next I suppose I'll reformulate the question and see if having the dots "evenly spaced" can lead to a formula.
  7. Jan 31, 2014 #6


    User Avatar
    Science Advisor

    This is a classic problem known as the "missing area" problem.

    If you place n dots around a circle NOT equally spaced, but so that the chords divide the interior of the circle into the maximum possible number of areas, the number of areas is given by
    [tex]\sum_{i=0}^4 \begin{pmatrix}n \\ i\end{pmatrix}[/tex]
    where [itex]\begin{pmatrix}n \\ i\end{pmatrix}[/itex] is 0 if i> n.

    If n 0, 1, 2, 3, or 4, that gives the binomial coefficient expansion of [itex](1+ 1)^n= 2^n[/itex]. For n= 5, it is missing only the last term, 1, so is [/itex]2^5- 1= 31[/itex]. For n= 6, it is missing the last two terms, 5(1)+ 1= 6, so is [itex]2^6- 6= 64- 6= 58[/itex].
  8. Jan 31, 2014 #7
    thanks! I'm gonna research this some more! Appreciate the help everyone!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Recreational problem involving circles, cords, and areas
  1. Area of a circle (Replies: 2)