Question

Explain why it is not possible to have a semi-regular polyhedron in which exactly three faces, a triangle, a square and an octagon, meet at each vertex.
Any tips would be much appreciated.

Answer 1

Answer:

Simplest Reason: Angles at a Vertex

The simplest reason there are only 5 Platonic Solids is this:

cube 3 faces meet at vertex

At each vertex at least 3 faces meet (maybe more).

cube 3 times 90 degrees at vertex

When we add up the internal angles that meet at a vertex,  

it must be less than 360 degrees.

four sqaures make 360 degrees, but flat

Because at 360° the shape flattens out!

And, since a Platonic Solid's faces are all identical regular polygons, we get:

regular triangle  

A regular triangle has internal angles of 60°, so we can have:

3 triangles (3×60°=180°) meet

4 triangles (4×60°=240°) meet

or 5 triangles (5×60°=300°) meet

regular quadrilateral  

A square has internal angles of 90°, so there is only:

3 squares (3×90°=270°) meet

pentagon regular  

A regular pentagon has internal angles of 108°, so there is only:

3 pentagons (3×108°=324°) meet

hexagon  

A regular hexagon has internal angles of 120°, but 3×120°=360° which won't work because at 360° the shape flattens out.

So a regular pentagon is as far as we can go.

And this is the result:

At each vertex: Angles at Vertex  

(Less than 360°) Solid  

3 triangles meet 180° tetrahedron Tetrahedron

4 triangles meet 240° octahedron Octahedron

5 triangles meet 300° icosahedron Icosahedron

3 squares meet 270° cube Cube

3 pentagons meet 324° dodecahedron Dodecahedron

Anything else has 360° or more at a vertex, which is impossible. Example: 4 regular pentagons (4×108° = 432°) won't work. And 3 regular hexagons (3×120° = 360°) won't work either.

And that is the simplest reason.

Step-by-step explanation: