Number Diagonals of Polygon

One of the easiest topics in plain Geometry is to count the number of diagonals in a convex polygon. You might familiar already or even memorized the said formula. So let me show you the technique how to derive it. This tutorial is made with two main reasons. First is to educate and second is to teach my own technique in derivation.

We will present two derivations, using plain geometry and common sense and the second way is using Combinatorics technique and again a small common sense.


By Plane Geometry:

diagonals of polygon

Consider the figure above of a convex heptagon. Each point can make 4 diagonals. Only four diagonals because we can’t consider the segment made by connecting point A to B and point A to C. Since there are 7 points, there must be 7×4=28 diagonals are there in heptagon.

But, try to draw the diagonals starting from point E like shown below.

diagonals of polygon2

We can basically draw again 4 diagonals but there is 1 common diagonal connecting A to E and E to A which should be counted only once. Doing the same process, we were able to figure out that each point has 1 common diagonal that should be counted once. Since we already calculated the number of diagonals previously, we can eliminate the error of counting one diagonal twice by dividing the final answer by 2. So instead of 28, the number of diagonals that can be drawn in convex heptagon is 14.

For n-gon:

Now, consider a convex n-gon. Since a heptagon has 7 sides and 7 vertices, n-gon also has n number of vertices. Now each point of n-gon can make n-3 diagonals. That is because the segments connecting the point and the point right beside it can’t be considered as diagonal.

So there must be n(n-3) diagonals that can be drawn from n-gon however remember that we counted each diagonal twice. By dividing n(n-3) by 2 we can eliminate counting errors. Thus the formula for taking the maximum number of diagonals from convex n-gon is


Derivation using basic Combinatorics with small common sense:

Consider a convex polygon like the figure shown below.

diagonals of polygon 3


Removing the segments connecting the points we have the figure in 2, a scattered points.

In Combinatorics, we can connect the points in nC2 ways from n number of points. But we also need to subtract the outermost segment that  we took out from figure 1 and the rest must be the number of diagonals. That is where the small common sense can be used.

Expressing that in equation we have,









 I hope you learned something from that. 😀



Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

Latest posts by Dan (see all)

You may also like...

20 Responses

  1. Hey, thanks for the article.Really thank you! Want more.

  2. Jame says:

    I like what you guys are up to. This type of clever work and coverage! Keep up the excellent work guys, I’ve included you blogroll.

  3. Yesterday, while I was at work, my sister stole my apple ipad and tested to see if it can survive a 30 foot drop, just so she can be a youtube sensation. My iPad is now broken and she has 83 views. I know this is entirely off topic but I had to share it with someone!

  4. this site says:

    l57211 I think this is a real great blog. Really Great.

  5. 758270 879369U never get what u expect u only get what u inspect 490024

  6. 239575 560430I dont leave a great deal of comments on a great deal of blogs each week but i felt i had to here. A hard-hitting post. 774564

  7. Spot on with this write-up, I absolutely feel this web site needs
    a lot more attention. I’ll probably be back again to read more, thanks for the info!

  8. With havin so much content and articles do you ever run into any problems of plagorism or copyright infringement?

    My blog has a lot of exclusive content I’ve either created myself or
    outsourced but it appears a lot of it is popping it up all over the web without my agreement.
    Do you know any solutions to help prevent content
    from being stolen? I’d definitely appreciate it.

  9. Say, you got a nice blog post.Really looking forward to read more.

  10. Hi there it’s me, I am also visiting this website daily, this website is
    truly nice and the visitors are in fact sharing pleasant thoughts.

  11. Wow, amazing blog format! How lengthy have you been blogging for?
    you made running a blog look easy. The entire look of your site is magnificent, as smartly as the content!

  12. of course like your web site however you need to test the
    spelling on quite a few of your posts. A number of them are
    rife with spelling problems and I in finding it very bothersome to tell the truth then again I’ll surely come again again.

  13. Enid says:

    I’m now not certain where you are getting your info, but good
    topic. I must spend some time studying more or working out more.
    Thanks for fantastic information I used to be in search of this information for my

  14. Howdy, i read your blog occasionally and i own a similar one and i was
    just curious if you get a lot of spam remarks? If so how do you stop it, any plugin or anything you can suggest?
    I get so much lately it’s driving me mad so any assistance is
    very much appreciated.

  15. read this says:

    I really liked your post. Great.

  16. Sweet blog! I found it while surfing around on Yahoo News.
    Do you have any tips on how to get listed in Yahoo News?
    I’ve been trying for a while but I never seem to get there!
    Many thanks

  17. van hire says:

    Hello Dear, are you truly visiting this web page regularly, if so after that you will
    without doubt obtain nice experience.

  18. equipments says:

    443689 171482Hi there, just became aware of your weblog by way of Google, and identified that it is genuinely informative. Im gonna watch out for brussels. Ill be grateful in case you continue this in future. A lot of men and women is going to be benefited from your writing. Cheers! 113298

  19. Hello, after reading this remarkable post i am also delighted to share my knowledge here with colleagues.

  20. says:

    We are a group of volunteers and starting a brand new scheme
    in our community. Your website offered us with useful information to work on. You’ve performed
    an impressive process and our whole neighborhood might
    be thankful to you.

Leave a Reply

Your email address will not be published. Required fields are marked *