Formulas and Tables

Elementary Geometry

Platonic Solids

Edge of a regular polyhedron: \(a\)
Volume of a polyhedron: \(V\)
Surface area: \(S\)

Radius of the inscribed sphere: \(r\)
Radius of the circumscribed sphere: \(R\)

  1. A polyhedron is a three-dimensional convex figure with flat faces and straight edges. If the faces are equal regular polygons, then the polyhedron is also called regular. The regular polyhedra have been known since deep antiquity. For example, the neolithic people of Scotland were able to create small stone balls representing the convex polyhedra.
Convex polyhedra made by the neolithic people of Scotland

Later, the Greek philosopher and mathematician Plato \(\left({428/427 \text{ BC}}\right.\) – \(\left.{348/347 \text{ BC}}\right)\) described in detail the regular polyhedra and identified five possible types of the solids (they are also called Platonic solids). The regular polyhedra include the regular tetrahedron, cube, octahedron, icosahedron and dodecahedron.

  1. Basic properties of platonic solids
Basic properties of platonic solids
  1. An octahedron is a regular polyhedron with \(8\) faces in the form of an equilateral triangle.
Octahedron
  1. Radius of a sphere inscribed in an octahedron
    \(r = {\large\frac{{a\sqrt 6 }}{6}\normalsize}\)
  2. Radius of a sphere circumscribed around an octahedron
    \(R = {\large\frac{{a\sqrt 2 }}{2}\normalsize}\)
  3. Surface area of an octahedron
    \(S = 2{a^2}\sqrt 3 \)
  4. Volume of an octahedron
    \(V = {\large\frac{{{a^3}\sqrt 2 }}{3}\normalsize}\)
  5. An icosahedron is a regular polyhedron with \(20\) faces having the form of an equilateral triangle.
Icosahedron
  1. Radius of a sphere inscribed in an icosahedron
    \(r = {\large\frac{{a\sqrt 3 \left( {3 + \sqrt 5 } \right)}}{{12}}\normalsize}\)
  2. Radius of a sphere circumscribed around an icosahedron
    \(R = {\large\frac{a}{4}\normalsize}\sqrt {2\left( {5 + \sqrt 5 } \right)} \)
  3. Surface area of an icosahedron
    \(S = 5{a^2}\sqrt 3 \)
  4. Volume of an icosahedron
    \(V = {\large\frac{{5{a^3}\left( {3 + \sqrt 5 } \right)}}{{12}}\normalsize}\)
  5. A dodecahedron is a regular polygon with \(12\) faces, which have the form of a regular pentagon.
Dodecahedron
  1. Radius of a sphere inscribed in a dodecahedron
    \(r =\) \({\large\frac{a}{2}\normalsize}\sqrt {10\left( {25 + 11\sqrt 5 } \right)} \)
  2. Radius of a sphere circumscribed around a dodecahedron
    \(R = {\large\frac{{a\sqrt 3 \left( {1 + \sqrt 5 } \right)}}{4}\normalsize}\)
  3. Surface area of a dodecahedron
    \(S =\) \(3{a^2}\sqrt {5\left( {5 + 2\sqrt 5 } \right)} \)
  4. Volume of a dodecahedron
    \(V = {\large\frac{{{a^3}\left( {15 + 7\sqrt 5 } \right)}}{4}\normalsize}\)