if every vertex of a regular pentagon is connected to every other vertex, how many triangles are formed?

Answer 1

There are 35 triangles. Count them this way: All triangles whose vertices are vertices of the pentagon. There are exactly 10 of hese, one for every different set of three vertices. Next count the triangles using exactly two adjacent vertices of the pentagon, plus one vertex inside the figure. Each pair of adjacent vertices has three such triangles, so we get 15 more. Now count the triangles which use two NON-adjacent vertices of the pentagon plus an inside vertex. There is just one triangle for each such pair, and there are five pairs, so 5 more triangles. Now count the triangles using one pentagon vertex and two inside vertices. There is only one for each vertex, so 5 more. There are NO triangles using only inside vertices, so we are done, and it adds up to 35.

I copied and pasted this directly from http://mathforum.org/library/drmath/view/54888.html

Answer 2

Vertices Of A Pentagon

Answer 3

10

Answer 4

11

Answer 5

The best way to solve this problem without even drawing the figure is by combination. The formula for combination is nCr=n!/[(n-r)!r!] Since the polygon is a pentagon, then n=5. Then, r=3 because a triangle has 3 sides. 5C3=5!/[((5-3)!)(2!)] 5C3=[5*4*3*2*1]/[(2*1)*(3*2*1)] 3C5=120/[2*6] 3C5=120/12 3C5=10 triangles

Answer 6

If we connect all vertices of a regular N-sided polygon we obtain a figure with N*(Nc2) = N*N (N - 1) / 2 triangles

So, for N=8
8*(8c2) = 8*8*(8 - 1) / 2 = 8*7/2 = 224.

these traingles are total in number and some of them can be formed as result of merging two or more traigles.

All the best

Answer 7

Vertex Of A Pentagon