1. ## Geometry

How many triangles are there in a polygon, if the polygon has 44 diagonals?

2. Originally Posted by sureshrju
How many triangles are there in a polygon, if the polygon has 44 diagonals?
Try to draw out the diagonals for the first couple of polygons. See if you can deduce a pattern in the number of triangles.

3. Hello Arcketer,

Draw the first couple of polygon in the sense? I can not understand

4. Hello, sureshrju!

Please check the wording of the problem . . .

How many triangles are there in a polygon, if the polygon has 44 diagonals?
A convex polygon with $n$ sides has: . $\frac{n(n-3)}{2}$ diagonals.

. . But . $\frac{n(n-3)}{2} \:=\:44$ .has no integral solutions. . . . . . wrong!

I did some very bad algebra . . . sorry!

5. Hi Saroban,

There is any polygon of concave? My friend ask this question to me, So i can not check the question again, i asked him about the question he said it is the correct one.

n(n-3)/2=44
n(n-3)=88
n^2-3n=88
n^2-3n-88=0
Then, (n+8)(n-11)=0
n=-8 or n=11
Since n denotes side, it can not be negative. So, n=11.
So, it is a polygon of 11 sides.

Can you please help me in finding out the number of triangles in the 11 sided polygon?

6. In fact $\frac{n(n-3)}{2}=44$ does have integral solutions: $11~\&~-8$.
Thus we have 11 vertices.
The number of triangles on 11 non-collinear points is $\binom{11}{3}$.
If you are asking for the total number of triangle formed by the diagonals and/or sides that is a different matter.

7. Hello Plato,

Is there any method to find the number of triangles formed by the sides and diagonals?

8. Originally Posted by sureshrju
Hello Plato,
Is there any method to find the number of triangles formed by the sides and diagonals?
That is the number I gave you: $\binom{11}{3}$.