Question

How many triangles are formed in a regular polygon with 13 sides?

Answer 1

A triangle can be formed by joining any 3 of the 13 vertices.

so the total number of triangle is C(13,3) = 13!/(3!10!) = 286

Answer 2

ok well that answer is not one of my choices. so something is wrong

Answer 3

The number of diagonals in the first series of polygons are

Number of sides
n = 3....4....5....6....7....8
Number of diagonals
N = 0....2....5....9...14...20
1st Difference
......2....3....4....5....6
2ndDifference
........1....1....1.....1

We therefore, have a finite difference sequence with the 2nd differences constant at 1. This means that the general expression for the number of diagonals in any n-gon is of the form N = an^2 + bn + c.

Using the data, we can write
a(3^2) + b(3) + c = 0 or 9a + 3b + c = 0
a(4^2) + b(4) + c = 2 or 16a + 4b + c = 2
a(5^2) + b(5) + c = 5 or 25a + 5b + c = 5

Solving this set of equations leads us to a = 1/2, b = -3/2, and c = 0 resulting in N = n^2/2 - 3n/2 = n(n - 3)/2.

Therefore, with 13 sides,
N = (13)^2/2 - 3(13)/2 = 65.

Answer 4

How many triangles are formed in a regular polygon with 13 sides?