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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 ...
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;
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 ...
Proposition one states: The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle. Any circle with a circumference c and a radius r is equal in area with a right triangle with the two legs being c and r.
However, it is constructible using neusis, or an angle trisector. The following is an animation from a neusis construction of a regular tridecagon with radius of circumcircle ¯ =, according to Andrew M. Gleason, [1] based on the angle trisection by means of the Tomahawk (light blue).
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 ...
A regular hendecagon is represented by Schläfli symbol {11}.. A regular hendecagon has internal angles of 147. 27 degrees (=147 degrees). [5] The area of a regular hendecagon with side length a is given by [2]