Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.4 AMC 10/12 Find the answer

A rhombic dodecahedron is a solid with $12$ congruent rhombus faces. At every vertex, $3$ or $4$ edges meet, depending on the vertex. How many vertices have exactly $3$ edges meet?

Pick one

Solution

Note Euler's formula where $\text{Vertices}+\text{Faces}-\text{Edges}=2$ . There are $12$ faces and the number of edges is $24$ because there are 12 faces each with four edges and each edge is shared by two faces. Now we know that there are $14$ vertices on the figure. Now note that the sum of the degrees of all the points is twice the number of edges. Let $x=$ the amount of vertices with $3$ edges. Now we know $\frac{3x+4(14-x)}{2}=24$ . Solving this system of equations gives $x = 8$ so the answer is $\boxed{\textbf{(D) }8}$ .
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