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Regular triacontatetragon  

A regular triacontatetragon


Type  Regular polygon 
Edges and vertices  34 
Schläfli symbol  {34}, t{17} 
Coxeter diagram  
Symmetry group  Dihedral (D_{34}), order 2×34 
Internal angle (degrees)  ≈169.41° 
Dual polygon  Self 
Properties  Convex, cyclic, equilateral, isogonal, isotoxal 
In geometry, a triacontatetragon or triacontakaitetragon is a thirtyfoursided polygon or 34gon.^{[1]} The sum of any triacontatetragon's interior angles is 5760 degrees.
A regular triacontatetragon is represented by Schläfli symbol {34} and can also be constructed as a truncated 17gon, t{17}, which alternates two types of edges.
One interior angle in a regular triacontatetragon is (2880/17)°, meaning that one exterior angle would be (180/17)°.
The area of a regular triacontatetragon is (with t = edge length)
and its inradius is
The factor is a root of the equation .
The circumradius of a regular triacontatetragon is
As 34 = 2 × 17 and 17 is a Fermat prime, a regular triacontatetragon is constructible using a compass and straightedge.^{[2]}^{[3]}^{[4]} As a truncated 17gon, it can be constructed by an edgebisection of a regular 17gon. This means that the values of and may be expressed in terms of nested radicals.
A triacontatetragram is a 34sided star polygon. There are seven regular forms given by Schläfli symbols {34/3}, {34/5}, {34/7}, {34/9}, {34/11}, {34/13}, and {34/15}, and nine compound star figures with the same vertex configuration.
{34/3} 
{34/5} 
{34/7} 
{34/9} 
{34/11} 
{34/13} 
{34/15} 
Many isogonal triacontatetragrams can also be constructed as deeper truncations of the regular heptadecagon {17} and heptadecagrams {17/2}, {17/3}, {17/4}, {17/5}, {17/6}, {17/7}, and {17/8}. These also create eight quasitruncations: t{17/9} = {34/9}, t{17/10} = {34/10}, t{17/11} = {34/11}, t{17/12} = {34/12}, t{17/13} = {34/13}, t{17/14} = {34/14}, t{17/15} = {34/15}, and t{17/16} = {34/16}. Some of the isogonal triacontatetragrams are depicted below, as a truncation sequence with endpoints t{17}={34} and t{17/16}={34/16}.^{[5]}
t{17}={34} 
t{17/16}={34/16} 