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A skew zig-zag dodecagon has vertices alternating between two parallel planes. A regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a hexagonal antiprism with the same D 5d, [2 +,10] symmetry, order 20. The dodecagrammic ...
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 ...
The area within a circle is equal to the radius multiplied by half the circumference, or A = r x C /2 = r x r x π.. Liu Hui argued: "Multiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, to yield the area of a dodecagon; if we cut a hexagon into a dodecagon, multiply its side by its radius, then again multiply by six, we get the area of a 24-gon; the ...
The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes; The volume of a sphere is 4 times that of a cone having a base of the same radius and height equal to this radius; The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter;
The volume is computed as F times the volume of the pyramid whose base is a regular p-gon and whose height is the inradius r. That is, =. The following table lists the various radii of the Platonic solids together with their surface area and volume.
the radius of the sphere passing through the eight order three vertices is exactly equal to the length of the sides: = The surface area A and the volume V of the rhombic dodecahedron with edge length a are: [ 4 ] A = 8 2 a 2 ≈ 11.314 a 2 , V = 16 3 9 a 3 ≈ 3.079 a 3 . {\displaystyle {\begin{aligned}A&=8{\sqrt {2}}a^{2}&\approx 11.314a^{2 ...
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 following formulae for circumradius, and height , surface area, and volume may be applied if all faces are regular with edge length : [4] =, = +, = + + (+), = +. It has an axis of symmetry passing through the center of both top and base, which is symmetrical by rotating around it at one-, two-, three-, and four-fifth of a full-turn angle.