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Icosahedron:
There are 3 possible lengths for the sides of the triangles.
The first length is equal to the edge of the icosahedron, and there are 20 triangles of this size, equal to the 20 faces of the icosahedron.
The second length is equal to the distance between two points that are two edges away from each other. If we consider a single vertex, we can count five triangles that include this vertex. There are twelve vertices, and if we multiply, 12*5, we will count each triangle three times. 12*5/3 is 20, so there are 20 triangles of this size.
The third distance is the distance across the icosahedron. No triangles can be formed of this size. The total stays at 40 triangles.
Dodecahedron:
There are 5 possible lengths of sides for the triangles. Like with the icosahedron, each length is equal to the distance between two points that are n edges away from each other, where n is 1 through 5. For n=1, 4, and 5, there are no triangles formed.
For n=2, there exist 3 triangles that include any given vertex, and this time there are 20 vertices. 20*3/3 is 20, so there are 20 triangles of this size.
For n=3, there exist 6 triangles that include any given vertex. 20*6/3 is 40, so there are 40 triangles of this size. There are 60 triangles total. |
