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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 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);
The apothem a can be used to find the area of any regular n-sided polygon of side length s according to the following formula, which also states that the area is equal to the apothem multiplied by half the perimeter since ns = p.
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 ...
As 24 = 2 3 × 3, a regular icositetragon is constructible using an angle trisector. [1] As a truncated dodecagon , it can be constructed by an edge- bisection of a regular dodecagon. Symmetry
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 ...
Draw the radius A B, bisect it in C—with an opening of the compasses equal to half the radius, upon A and C as centres describe the arcs C D I and A D—with the distance I D upon I describe the arc D O and draw the line C O, which will be the extent of one side of a hendecagon sufficiently exact for practice. On a unit circle:
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.