Question

A convex polyhedron has faces that consist of 30 squares, 20 triangles, and 12 pentagons. The polyhedron has 120 edges. How many vertices does it have?

Euler’s formula: V + F = E + 2

60
92
122
180

Answer 1

The number of vertices in the polyhedron can be found using Euler's formula:

V + F = E + 2

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

We are given that the polyhedron has 30 squares, 20 triangles, and 12 pentagons. Since each square has 4 sides, each triangle has 3 sides, and each pentagon has 5 sides, the total number of sides in the polyhedron is:

30 x 4 + 20 x 3 + 12 x 5 = 120 + 60 + 60 = 240

We are also given that the polyhedron has 120 edges, so:

E = 120

Finally, we can substitute these values into Euler's formula and solve for V:

V + F = E + 2

V + 30 + 20 + 12 = 120 + 2

V + 62 = 122

V = 60

Therefore, the polyhedron has 60 vertices. Answer: 60.