Find the number of vertices in a polyhedron if it contains 20 triangular faces and 12 pentagonal faces.
a. 30
b. 32
c. 60
d. 120
* Also please show me how you got your answer, thanks
Recall Euler's Formula:
V+F-E=2
and check out
https://en.wikipedia.org/wiki/Icosidodecahedron
To find the number of vertices in a polyhedron, we can use Euler's formula. Euler's formula states that for any convex polyhedron, the number of vertices (V), the number of edges (E), and the number of faces (F) are related by the equation V + F - E = 2.
Given that the polyhedron contains 20 triangular faces and 12 pentagonal faces, we can calculate the total number of faces by adding the number of triangular faces (20) and the number of pentagonal faces (12), resulting in a total of 32 faces (F = 20 + 12 = 32).
Now, we need to determine the number of edges. Since each triangular face has 3 edges and each pentagonal face has 5 edges, we multiply the number of triangular faces by 3 and the number of pentagonal faces by 5, and then sum them together. This gives us a total of 60 edges (E = 20 * 3 + 12 * 5 = 60).
Substituting the values into Euler's formula, we have V + 32 - 60 = 2. Solving for V, we get V = 30.
Therefore, the correct answer is option a. 30.