Polyhedra
Overview
|
Tetrahedron |
Octahedron |
Hexahedron |
Dodecahedron |
Icosahedron |
| Vertices |
4 |
6 (2 × 3) |
8 |
12 (4 × 3) |
12 |
| Edges |
6 |
12 |
12 |
30 |
30 |
| Faces |
4 |
8 |
6 |
20 |
20 (8 + 4 × 3) |
| Face shape |
Equilateral triangle |
Equilateral triangle |
Square |
Regular pentagon |
Equilateral triangle |
| Schläfli symbol |
{3, 3} |
{3, 4} |
{4, 3} |
{5, 3} |
{3, 5} |
| Wythoff symbol |
3 | 2 3 |
4 | 2 3 |
3 | 2 4 |
3 | 2 5 |
5 | 2 3 |
| C&R symbol |
33 |
34 |
43 |
53 |
35 |
| Vertex configuration |
3.3.3 |
3.3.3.3 |
4.4.4 |
5.5.5 |
3.3.3.3.3 |
Parameters
Entities
| Parameter |
Symbol |
Name |
Description |
p |
\(P\), \(V\) |
Vertex |
|
e |
\(E\) |
Edge |
|
f |
\(F\) |
Face |
|
s |
\(S\) |
Solid |
|
Properties
| Parameter |
Symbol |
Name |
Description |
r |
\(r\) |
Radius |
|
t |
\(\theta\) |
Angle |
|
h |
\(h\) |
Height |
|
l |
\(l\) |
Length |
|
a |
\(A\) |
Area |
|
v |
\(V\) |
Volume |
|
Combined
| Parameter |
Symbol |
Name |
Description |
p_c |
\(x\) |
Vertex coordinate |
|
e_l |
\(a\) |
Edge length |
|
f_a |
\(A0\) |
Face area |
|
r_u |
\(R\), \(r_u\) |
Circumradius |
The circumradius of a cyclic polygon is a radius of the circle inside which the polygon can be inscribed. Similarly, the circumradius of a polyhedron is the radius of a circumsphere touching each of the polyhedron's vertices, if such a sphere exists. |
r_i |
\(r\), \(r_i\) |
Inradius |
The radius of a polygon's incircle or of a polyhedron's insphere. |
r_m |
\(\rho\), \(r_m\) |
Midradius |
The radius of the midsphere of a polyhedron, also called the interradius. |
s_a |
\(A\) |
Surface area |
|
s_v |
\(V\) |
Volume |
|
s_t |
\(\omega\) |
Solid Angle |
|
f_t_d |
\(a\) |
Dihedral Angle |
The dihedral angle is the angle between two planes. |
s_t_s |
\(\beta\) |
Subtended Angle |
Subtended by an edge from the center for the Platonic solids. |
p_t |
|
Vertex Angle |
|
https://www.technologyuk.net/mathematics/geometry/platonic-solids.shtml