Tetracontagon
Regular tetracontagon  

A regular tetracontagon  
Type  Regular polygon 
Edges and vertices  40 
Schläfli symbol  {40}, t{20}, tt{10}, ttt{5} 
Coxeter diagram 

Symmetry group  Dihedral (D_{40}), order 2×40 
Internal angle (degrees)  171° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a tetracontagon or tessaracontagon is a fortysided polygon or 40gon.^{[1]}^{[2]} The sum of any tetracontagon's interior angles is 6840 degrees.
Regular tetracontagon
A regular tetracontagon is represented by Schläfli symbol {40} and can also be constructed as a truncated icosagon, t{20}, which alternates two types of edges. Furthermore, it can also be constructed as a twicetruncated decagon, tt{10}, or a thricetruncated pentagon, ttt{5}.
One interior angle in a regular tetracontagon is 171°, meaning that one exterior angle would be 9°.
The area of a regular tetracontagon is (with t = edge length)
and its inradius is
The factor is a root of the octic equation .
The circumradius of a regular tetracontagon is
As 40 = 2^{3} × 5, a regular tetracontagon is constructible using a compass and straightedge.^{[3]} As a truncated icosagon, it can be constructed by an edgebisection of a regular icosagon. This means that the values of and may be expressed in radicals as follows:
Construction of a regular tetracontagon
Circumcircle is given
 Construct first the side length JE_{1} of a pentagon.
 Transfer this on the circumcircle, there arises the intersection E_{39}.
 Connect the point E_{39} with the central point M, there arises the angle E_{39}ME_{1} with 72°.
 Halve the angle E_{39}ME_{1}, there arise the intersection E_{40} and the angle E_{40}ME_{1} with 9°.
 Connect the point E_{1} with the point E_{40}, there arises the first side length a of the tetracontagon.
 Finally you transfer the segment E_{1}E_{40} (side length a) repeatedly counterclockwise on the circumcircle until arises a regular tetracontagon.
The golden ratio
Side length is given
 Draw a segment E_{40}E_{1} whose length is the given side length a of the tetracontagon.
 Extend the segment E_{40}E_{1} by more than two times.
 Draw each a circular arc about the points E_{1} and E_{40}, there arise the intersections A and B.
 Draw a vertical straight line from point B through point A.
 Draw a parallel line too the segment AB from the point E_{1} to the circular arc, there arises the intersection D.
 Draw a circle arc about the point C with the radius CD till to the extension of the side length, there arises the intersection F.
 Draw a circle arc about the point E_{40} with the radius E_{40}F till to the vertical straight line, there arises the intersection G and the angle E_{40}GE_{1} with 36°.
 Draw a circle arc about the point G with radius E_{40}G till to the vertical straight line, there arises the intersection H and the angle E_{40}HE_{1} with 18°.
 Draw a circle arc about the point H with radius E_{40}H till to the vertical straight line, there arises the central point M of the circumcircle and the angle E_{40}ME_{1} with 9°.
 Draw around the central point M with radius E_{40}M the circumcircle of the tetracontagon.
 Finally transfer the segment E_{40}E_{1} (side length a) repeatedly counterclockwise on the circumcircle until to arises a regular tetracontagon.
The golden ratio
Symmetry
The regular tetracontagon has Dih_{40} dihedral symmetry, order 80, represented by 40 lines of reflection. Dih_{40} has 7 dihedral subgroups: (Dih_{20}, Dih_{10}, Dih_{5}), and (Dih_{8}, Dih_{4}, Dih_{2}, Dih_{1}). It also has eight more cyclic symmetries as subgroups: (Z_{40}, Z_{20}, Z_{10}, Z_{5}), and (Z_{8}, Z_{4}, Z_{2}, Z_{1}), with Z_{n} representing π/n radian rotational symmetry.
John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[4]} He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry. a1 labels no symmetry.
These lower symmetries allows degrees of freedoms in defining irregular tetracontagons. Only the g40 subgroup has no degrees of freedom but can seen as directed edges.
Dissection
regular 
Isotoxal 
Coxeter states that every zonogon (a 2mgon whose opposite sides are parallel and of equal length) can be dissected into m(m1)/2 parallelograms. These tilings are contained as subsets of vertices, edges and faces in orthogonal projections mcubes^{[5]} In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular tetracontagon, m=20, and it can be divided into 190: 10 squares and 9 sets of 20 rhombs. This decomposition is based on a Petrie polygon projection of a 20cube.
Tetracontagram
A tetracontagram is a 40sided star polygon. There are seven regular forms given by Schläfli symbols {40/3}, {40/7}, {40/9}, {40/11}, {40/13}, {40/17}, and {40/19}, and 12 compound star figures with the same vertex configuration.
Picture  {40/3} 
{40/7} 
{40/9} 
{40/11} 
{40/13} 
{40/17} 
{40/19} 

Interior angle  153°  117°  99°  81°  63°  27°  9° 
Picture  {40/2}=2{20} 
{40/4}=4{10} 
{40/5}=5{8} 
{40/6}=2{20/3} 
{40/8}=8{5} 
{40/10}=10{4} 

Interior angle  162°  144°  135°  126°  108°  90° 
Picture  {40/12}=4{10/3} 
{40/14}=2{20/7} 
{40/15}=5{8/3} 
{40/16}=8{5/2} 
{40/18}=2{20/9} 
{40/20}=20{2} 
Interior angle  72°  54°  45°  36°  18°  0° 
Many isogonal tetracontagrams can also be constructed as deeper truncations of the regular icosagon {20} and icosagrams {20/3}, {20/7}, and {20/9}. These also create four quasitruncations: t{20/11}={40/11}, t{20/13}={40/13}, t{20/17}={40/17}, and t{20/19}={40/19}. Some of the isogonal tetracontagrams are depicted below, as a truncation sequence with endpoints t{20}={40} and t{20/19}={40/19}.^{[6]}
t{20}={40} 

t{20/19}={40/19} 
References
 ↑ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 165, ISBN 9781438109572 .
 ↑ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
 ↑ Constructible Polygon
 ↑ The Symmetries of Things, Chapter 20
 ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
 ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum