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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.
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 ...
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]
The area bounded by one spiral rotation and a line is 1/3 that of the circle having a radius equal to the line segment length; Use of the method of exhaustion also led to the successful evaluation of an infinite geometric series (for the first time);
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 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 ...
The 120-cell of edge-length 2 and long radius φ 2 √ 8 ≈ 7.405 given by Coxeter [3] can be constructed in this manner just outside a 600-cell of long radius φ 4 and edge-length φ 3. Therefore, the unit-radius 120-cell can be constructed from its predecessor the unit-radius 600-cell in three reciprocation steps.
Comparison of sizes of regular polygons with the same edge length, from three to sixty sides. The size increases without bound as the number of sides approaches infinity. Of all n-gons with a given perimeter, the one with the largest area is regular. [10]