A polyhedron has $38$ faces (only squares and triangles) and $60$ edges. Find the ratio of square faces to triangle faces.

Question

A polyhedron consists of faces only of squares and equilateral triangles. Given that it has $38$ faces and $60$ edges, find the ratio of square faces to triangular faces.

What I tried so far:

I let $x$ be the number of squares and $y$ be the number of triangles. Then we know that $x+y=38$. However, I am having trouble using $x$ and $y$ to relate it to the $60$ since it is harder to visualize.

Can anybody help?

How many edges do the triangles have? The squares? How many times have you counted each edge? — Ross Millikan, Aug 19 '20 at 23:54

score 2 · Accepted Answer · answered Aug 19 '20 at 23:50

2

Note that each edge of the polyhedron is connected to exactly $2$ faces. Since triangles have $3$ edges and squares have $4$ edges, then $4x+3y=2\times 60$, which together with $x+y=38$ implies that $x=6$ and $y=32$.

answered Aug 19 '20 at 23:50

YiFan Tey

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A polyhedron has $38$ faces (only squares and triangles) and $60$ edges. Find the ratio of square faces to triangle faces.

1 Answers1