Why does a three-dimensional object consisting of regular pentagons have twelve surfaces?

I know because I constructed one, but I'd really like to know why.Why does a three-dimensional object consisting of regular pentagons have twelve surfaces?
Suppose that you are trying to make a solid with n pentagons. These will have a total of 5n sides but when you put them together you join pairs of sides to make an edge of the solid so it will have 5n/2 edges. The n pentaongs will have a total of 5n corners but when they are joined to make a solid three will combine at one vertex of the solid so it will have 5n/3 vertices. (You know that three must combine at a vertex because two would not make 360 deg. and four or more can't join together in the way that you want.) The n pentagons of course make the n faces of the solid.


If you put these figures into Euler's equation F + V - E = 2 mentioned above you get n + 5n/3 - 5n/2 = 2. When you solve this equation you get n = 12.Why does a three-dimensional object consisting of regular pentagons have twelve surfaces?
The same reason a square has six! Good question. But Sorry I haven't got a clue.
first, i don't know. but maybe this would be a place to start.


but, first we might look at a 3 dimensional object made out of triangles.


it can have, 4, or 8, or 20 sides.


and if it's squares, then it's 6 sides.


but back to triangles.


if the corner angles (in 1 vertex) add up to:


180, there's 4 sides.


240, there's 8 sides.


300, there's 20 sides.





for reference, 270=6, and 324=12 so we need to confine any analysis to triangles.
Not sure. All I know is that it's called a dodecahedron, and I'm not sure if it's spelt correctly.
It has to do with a formula (found by Euler), that, for any regular polyhedron F + V - E = 2, where F = faces, V = vertices, and E = edges. From that there is a way to prove that the number of sides for a polyhedron with three pentagons at each vertex must be 12. Unfortunately, I can't remember how that proof is done, but I'll bet someone here does...





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OK, thinking about this, I think it is all coming back to me. You need three more ratios to solve this.





First, the face vs edge ratio. Since you have only pentagons, each face has five edges, while each edge is part of two faces. So, E/F = 5/2.





Next, the edge vs vertex ratio. Since there are three faces meeting at each vertex, each vertex has three edges, while each edge has two vertices. So, E/V = 3/2.





Lastly, the face vs vertex ratio. Each vertex has three edges and each face has five vertices. So, F/V = 3/5.





Add Euler's equation to this, and you have a uniquely solvable system of equations: F= 12, V = 20, and E = 30.