Number of Triangles!
Consider a regular
-sided polygon. Calculate the number of triangles formed by connecting all pairs of vertices with line segments.
Note: We are looking for the number of triangles that are actually visible once all the line segments are drawn.
Also try:
The answer is 101745.
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1 solution
Number of triangle can be form from connecting all pairs of vertices in n -sided polygon is: ( 3 n ) + 4 ( 4 n ) + 5 ( 5 n ) + ( 6 n ) .
So, in 1 9 -sided polygon ( n = 1 9 ) :
= ( 3 1 9 ) + 4 ( 4 1 9 ) + 5 ( 5 1 9 ) + ( 6 1 9 )
= 9 6 9 + 1 5 5 0 4 + 5 8 1 4 0 + 2 7 1 3 2
= 1 0 1 7 4 5