Answer:
Explanation:
Euler's formula states that for any polyhedron with V vertices, E edges, and F faces, we have:
V - E + F = 2
Let's verify this formula for the other four Platonic solids:
1. Cube:
A cube has 8 vertices, 12 edges, and 6 faces (all of which are squares).
So, we have:
V = 8
E = 12
F = 6
Plugging these values into Euler's formula, we get:
8 - 12 + 6 = 2
which is true, so Euler's formula works for a cube.
2. Octahedron:
An octahedron has 6 vertices, 12 edges, and 8 faces (all of which are equilateral triangles).
So, we have:
V = 6
E = 12
F = 8
Plugging these values into Euler's formula, we get:
6 - 12 + 8 = 2
which is true, so Euler's formula works for an octahedron.
3. Dodecahedron:
A dodecahedron has 20 vertices, 30 edges, and 12 faces (all of which are regular pentagons).
So, we have:
V = 20
E = 30
F = 12
Plugging these values into Euler's formula, we get:
20 - 30 + 12 = 2
which is true, so Euler's formula works for a dodecahedron.
4. Icosahedron:
An icosahedron has 12 vertices, 30 edges, and 20 faces (all of which are equilateral triangles).
So, we have:
V = 12
E = 30
F = 20
Plugging these values into Euler's formula, we get:
12 - 30 + 20 = 2
which is true, so Euler's formula works for an icosahedron.
Therefore, we have shown that Euler's formula works for all five Platonic solids.