Hectogon
Regular hectogon | |
---|---|
A regular hectogon | |
Type | Regular polygon |
Edges and vertices | 100 |
Schläfli symbol |
{100} t{50} |
Coxeter diagram |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | Dihedral (D100), order 2×100 |
Internal angle (degrees) | 176.4° |
Dual polygon | self |
Properties | convex, cyclic, equilateral, isogonal, isotoxal |
In geometry, a hectogon or hecatontagon[1][2] is a hundred-sided polygon.[3][4] The sum of any hectogon's interior angles is 17640 degrees.
A regular hectogon is represented by Schläfli symbol {100} and can be constructed as a quasiregular truncated pentacontagon, t{50}, which alternates two types of edges.
Regular hectogon properties
One interior angle in a regular hectogon is 176.4°, meaning that one exterior angle would be 3.6°.
The area of a regular hectogon is (with t = edge length)
and its inradius is
The circumradius of a regular hectogon is
A regular hectogon is not constructible using a compass and straightedge,[5] and is not constructible even if the use of an angle trisector is allowed.[6]
Hectogram
A hectogram is an 100-sided star polygon. There are 19 regular forms[7] given by Schläfli symbols {100/3}, {100/7}, {100/9}, {100/11}, {100/13}, {100/17}, {100/19}, {100/21}, {100/23}, {100/27}, {100/29}, {100/31}, {100/33}, {100/37}, {100/39}, {100/41}, {100/43}, {100/47}, and {100/49}, as well as 30 regular star figures with the same vertex configuration.
Picture | ![]() {100/3} |
![]() {100/7} |
![]() {100/11} |
![]() {100/13} |
![]() {100/17} |
![]() {100/19} | |
---|---|---|---|---|---|---|---|
Interior angle | 169.2° | 154.8° | 140.4° | 133.2° | 118.8° | 111.6° | |
Picture | ![]() {100/21} |
![]() {100/23} |
![]() {100/27} |
![]() {100/29} |
![]() {100/31} |
![]() {100/37} |
|
104.4° | 97.2° | 82.8° | 75.6° | 68.4° | 46.8° | ||
Picture | ![]() {100/39} |
![]() {100/41} |
![]() {100/43} |
![]() {100/47} |
![]() {100/49} |
||
Interior angle | 39.6° | 32.4° | 25.2° | 10.8° | 3.6° |
References
- ↑
- ↑
- ↑ Gorini, Catherine A. (2009), The Facts on File Geometry Handbook, Infobase Publishing, p. 110, ISBN 9781438109572.
- ↑ The New Elements of Mathematics: Algebra and Geometry by Charles Sanders Peirce (1976), p.298
- ↑ Constructible Polygon
- ↑ http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf
- ↑ 19 = 50 cases - 1 (convex) - 10 (multiples of 5) - 25 (multiples of 2)+ 5 (multiples of 2 and 5)
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