Question 1181235
.
An icosahedron has 20 faces, all of which are equilateral triangles. How many vertices are there on a
regular icosahedron?
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<pre>
Use the Euler formula for convex polyhedrons

    F - E + V = 2      (1)


(as a reference, see this Wikipedia article https://en.wikipedia.org/wiki/Euler_characteristic ),

where F is the number of faces, E is the number of edges and V is the number of vertices.



The number of faces is given in the problem: F = 20.


The number of edges is three times the number of faces,  i.e. 3*20 = 60  as a first estimate.

But since every edge is a common side of two triangles, we should divide 60 by 2 to get the final estimate
of edges as 60/2 = 30.


So, the formula (1) is now

    20 - 30 + V = 2,


which gives  V = 2 - 20 + 30 = 12.


<U>ANSWER</U>.  The number of vertices of icosahedron is 12.
</pre>


On icosahedron, see the same Wikipedia article  https://en.wikipedia.org/wiki/Euler_characteristic 


It confirms that the number of vertices of a regular icosahedron is 12.


From the article, you can see how a regular icosahedron looks like.


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Solved.



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