A regular icosahedron is a three-dimensional solid composed of

BTGModeratorVI wrote: ↑

Wed May 13, 2020 10:58 am

A regular icosahedron is a three-dimensional solid composed of twenty equilateral triangular faces, with five faces meeting at each vertex. How many vertices does a regular icosahedron have?

A. 5
B. 12
C. 20
D. 33
E. 60

Answer: B
Source: Veritas Prep

Each triangle has 3 vertices, and there are 20 triangles in total.
So, the total number of triangle vertices = 3 x 20 = 60

Each vertex in the icosahedron is comprised of five triangle faces meeting at a point.
In other words, it takes 5 triangle vertices to make 1 icosahedron vertex .
We have 60 triangle vertices, so the TOTAL number of icosahedron vertices = 60/5 = 12

Answer: B

Cheers,
Brent

BTGModeratorVI wrote: ↑

Wed May 13, 2020 10:58 am

A regular icosahedron is a three-dimensional solid composed of twenty equilateral triangular faces, with five faces meeting at each vertex. How many vertices does a regular icosahedron have?

A. 5
B. 12
C. 20
D. 33
E. 60

Answer: B
Source: Veritas Prep

Solution:

We can divide a regular icosahedron into 3 sections: a top section with 5 triangular faces, a middle section with 10 triangle faces and a bottom section with 5 triangular faces (note: the bottom section is a symmetry of the top section).

The top section of 5 triangular faces has 6 vertices: one from the concurrency of the 5 faces and the other 5 from bases of the triangles. Since the bottom section is a symmetry of the top section, the bottom section also has 6 vertices. All the vertices of the triangles that are in the middle section that connects the base of the top section and the base of the bottom section are already accounted for since they either form the base of the top section or the base of the bottom section.

Therefore, there are 6 + 6 = 12 vertices in a regular icosahedron.

Alternate Solution:

20 triangles have 20 x 3 = 60 vertices in total. However, since five faces meet at each vertex, a vertex of the regular icosahedron appears five times in the 60 vertices. Therefore, there are 60/5 = 12 vertices.

Answer: B