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An icosahedron has 20 faces, all of which are equilateral triangles. How many vertices are there on a
regular icosahedron?
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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.
ANSWER. The number of vertices of icosahedron is 12.
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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