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Three squares of sides R can be cut and rearranged into a dodecagon of circumradius R, yielding a proof without words that its area is 3R 2. A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12.
Proposition 2: The area of circles is proportional to the square of their diameters. [3] Proposition 5: The volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases. [4] Proposition 10: The volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height. [5]
If the edge length of a regular dodecahedron is , the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices), the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces), and the midradius (one that touches the middle of each edge) are: [21] =, =, =. Given a regular dodecahedron ...
Variations can be parametrized by (a,b), where b and a depend on each other such that the tetrahedron defined by the four vertices of a face has volume zero, i.e. is a planar face. (1,1) is the rhombic solution. As a approaches 1 / 2 , b approaches infinity. It always holds that 1 / a + 1 / b = 2, with a, b > 1 / 2 .
The truncated dodecahedron is constructed from a regular dodecahedron by cutting all of its vertices off, a process known as truncation. [1] Alternatively, the truncated dodecahedron can be constructed by expansion: pushing away the edges of a regular dodecahedron, forming the pentagonal faces into decagonal faces, as well as the vertices into triangles. [2]
When an {n ⁄ d}-gon is truncated, it becomes a {2n ⁄ d}-gon. For example, a truncated pentagon {5 ⁄ 1} becomes a decagon {10 ⁄ 1}, so truncating a pentagram {5 ⁄ 2} becomes a doubly-wound pentagon {10 ⁄ 2} (the common factor between 10 and 2 mean we visit each vertex twice to complete the polygon).
The 120-cell whose coordinates are given above of long radius √ 8 = 2 √ 2 ≈ 2.828 and edge-length 2 / φ 2 = 3− √ 5 ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ 2, which is smaller than √ 8 in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell ...
Net. In geometry, a dodecahedral prism is a convex uniform 4-polytope.This 4-polytope has 14 polyhedral cells: 2 dodecahedra connected by 12 pentagonal prisms.It has 54 faces: 30 squares and 24 pentagons.