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Show that Euler's formula works for the other four Platonic solids: a cube, an octahedron, a dodecahedron, and an icosahedron.​

User UpaJah
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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.

User Vadchen
by
8.6k points
3 votes

Answer:

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.

User Studioromeo
by
8.6k points
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