A dodecahedron is a regular polyhedron,Whose faces are twelve regular pentagons. The simplest for constructing a regular polyhedron is a hexahedron or a cube, all the other polyhedra can be constructed by inscribing or describing them around it.
A dodecahedron can be constructed by describing it near a cube.
Construct a cube with edge length a. Calculate the length of the dodecahedron under construction using the formula: m = -a / 2 + av5 / 2, where a is the length of the edge of the cube.
On the SPRQ face, draw a line K1L1 connecting the middle of the edges. On this line, postpone a segment of length m equidistant from the edges of the cube. Through the ends of the segment, draw perpendiculars to the face of SPRQ.
Construct the pentagon ABCDE with the diagonals AC and BE. AB = BC = a. Calculate the height of the triangle ABC and designate it s = BN.
On perpendiculars, find the points, the distance from which to the midpoints of the edges is s, that is, LL1 = KK1 = s. Now connect the found points to the vertices of the cube.
Repeat constructions 2 and 4 for each face, as a result you will get the right polyhedron described near the cube - the dodecahedron.