Questions related to polyhedra and their properties.

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23
votes
4answers
14k views

Why are there 12 pentagons and 20 hexagons on a soccer ball?

Edge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron. Is there some insight into why the alternation of pentagons and hexagons yields an approximated ...
2
votes
1answer
430 views

Find the Polyhedron Enclosed by Multiple Faces

I have two faces ($S_1$ and $S_2$), each bounded by a series of Vertex $V$. These two faces may or may not intersect. In addition to that there are two vertical faces $S_3$ and $S_4$, that connect ...
7
votes
3answers
4k views

Tetrahedron inside a sphere

What's the largest regular tetrahedron (having side length $x$) you can fit inside a sphere with a unit radius?
8
votes
5answers
25k views

Height of a tetrahedron

How do I calculate the height of a regular tetrahedron having side length $1$ ? Just to be completely clear, by height I mean if you placed the shape on a table, how high up would the highest point ...
3
votes
1answer
352 views

Tetrahedron volume

How to calculate volume of tetrahedron given lengths of all it's edges?
15
votes
1answer
404 views

What property of certain regular polygons allows them to be faces of the Platonic Solids?

It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes. What property of those regular ...