The total number of diagonals in a decagon is..

I need a general relation which will hold true for all polygons!

Question

The total number of diagonals in a decagon is..

I need a general relation which will hold true for all polygons!

accessdenied · Answer

I would draw a diagram in this case and just count the number of unique diagonals from each point.  As for the relation, you may need to consider a few more cases or otherwise use intuition on how these diagonals are counted.

ujjwal · Answer

Ah, it says the answer is 110.
I am scared to count.

accessdenied · Answer

Hmm... that's a lot for a decagon, actually.
I'm only getting 35.

saifoo.khan · Answer

access is right.

accessdenied · Answer

110, which is 11*10 is like, 11 for each point, and there's not even that many points to go to, let alone having unique points.   D:

ujjwal · Answer

good point!

ujjwal · Answer

Are you sure, It's only 35. ?

anonymous · Answer

You could think of it like this.  There are 10 points in a decagon, and you need 2 to make a diagonal. So how many ways are there to choose 2 distinct points from 10?  Thats C(10,2) which is 45.  Now take away the 10 sides of the decagon.  That leaves you with 35.

anonymous · Answer

Or you could think of it like this.  Lay 10 points around in a circle and start at one of them.  Draw a line from that starting point to each of the other points.  You will draw 9 lines.  Then go the another point and draw lines to the other points that arent connected already.  You will draw 8 lines.  Then 7, 6, so on.  This gives:$$9+8+7+\cdots +2+1 = 45$$ Again subtract 10 for the sides to leave the diagonals.

ujjwal · Answer

@joemath314159 Does this method hold true for other polygons like pentagon and hexagon?

anonymous · Answer

It should.  Say for a hexagon.  There are 6 points.  C(6,2) = 15.  take away 6 for the sides, so there are 9 diagonals.

ujjwal · Answer

|dw:1339804971418:dw| we have 4 diagonals here. But c(5,2)-5=5

accessdenied · Answer

|dw:1339805080366:dw|

anonymous · Answer

|dw:1339805085317:dw|

ujjwal · Answer

Oh, i missed one. Thanks @joemath314159 and @AccessDenied .

accessdenied · Answer

You're welcome.  Using the combination idea, you can make / simplify a generalized formula too. :D

ujjwal · Answer

yeah, the one involving the idea of combination is very easy and i understood it.

anonymous · Answer

The logical argument is whats important.  How do you make a diagonal in a polygon? You have to choose 2 points and connect them.  So the geometric problem of drawing diagonals becomes a combinatorial problem instead, which requires no pictures or things of the like to solve.  This is why math is great :)