4147:
807:
436:
29:
4301:
235:
262:: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. This condition is true for polygons in any geometry, not just Euclidean.
2624:
2650:
Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon or to the area of one or more simple polygons having the same outline as the self-intersecting one. In the case of the cross-quadrilateral,
4414:
The imaging system calls up the structure of polygons needed for the scene to be created from the database. This is transferred to active memory and finally, to the display system (screen, TV monitors etc.) so that the scene can be viewed. During this process, the imaging system renders polygons in
2414:
139:
is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a
1389:
2646:
of the region. For example, the central convex pentagon in the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole
1178:
4217:
5010:
2456:
5679:
2093:
3004:
2832:
4415:
correct perspective ready for transmission of the processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation.
3956:
As with René Descartes's example of the chiliagon, the million-sided polygon has been used as an illustration of a well-defined concept that cannot be visualised. The megagon is also used as an illustration of the convergence of
769:, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an
392:
The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral. A non-convex regular polygon is called a
940:
2292:
1192:
1002:
3220:
3139:
5676:
1536:-axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in
1614:
569:
3417:
Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians. Some special polygons also have their own names; for example the
2258:
698:
747:
604:
283:: the whole interior is visible from at least one point, without crossing any edge. The polygon must be simple, and may be convex or concave. All convex polygons are star-shaped.
4913:
265:
Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon.
2619:{\displaystyle A={\frac {ns^{2}}{4}}\cot {\frac {\pi }{n}}={\frac {ns^{2}}{4}}\cot {\frac {\alpha }{n-2}}=n\cdot \sin {\frac {\alpha }{n-2}}\cdot \cos {\frac {\alpha }{n-2}}.}
4472:
2156:
641:
3980:
To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher and was used by
1520:
1471:
1609:
32:
Some polygons of different kinds: open (excluding its boundary), boundary only (excluding interior), closed (including both boundary and interior), and self-intersecting.
3331:
is a three-dimensional solid bounded by flat polygonal faces, analogous to a polygon in two dimensions. The corresponding shapes in four or higher dimensions are called
2838:
2666:
2448:
1425:
2105:
gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1.
4698:
4146:
3027:
3375:, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a
5828:, Provides an interactive Java investigation that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons
2642:
Using the formulas for simple polygons, we allow that particular regions within the polygon may have their area multiplied by a factor which we call the
6959:
5660:
5416:
5228:
5068:
4672:
2409:{\displaystyle A=R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=R^{2}\cdot n\cdot \sin {\frac {\pi }{n}}\cdot \cos {\frac {\pi }{n}}}
2660:
Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are
817:
3510:
The simplest polygon which can cross itself; the simplest polygon which can be concave; the simplest polygon which can be non-cyclic. Can
5667:
Reprint of original 1921 publication with corrected errata. Heath uses the
Latinized spelling "Aristophonus" for the vase painter's name.
1384:{\displaystyle 16A^{2}=\sum _{i=0}^{n-1}\sum _{j=0}^{n-1}{\begin{vmatrix}Q_{i,j}&Q_{i,j+1}\\Q_{i+1,j}&Q_{i+1,j+1}\end{vmatrix}},}
6394:
2168:
The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides
5263:
Arthur
Baragar (2002) Constructions Using a Compass and Twice-Notched Straightedge, The American Mathematical Monthly, 109:2, 151–164,
3414:
Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon.
5853:
1173:{\displaystyle A={\frac {1}{2}}\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})\quad {\text{where }}x_{n}=x_{0}{\text{ and }}y_{n}=y_{0},}
286:
172:
3242:, a polygon having only two sides and two corners, which is impossible in a flat plane. Spherical polygons play an important role in
6981:
5633:
5610:
5572:
5549:
5526:
5503:
5465:
5442:
4340:
3272:
is an infinite sequence of sides and angles, which is not closed but has no ends because it extends indefinitely in both directions.
3286:
is an area-connected or multiply-connected planar polygon with one external boundary and one or more interior boundaries (holes).
3145:
3064:
4833:
3925:
765:-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full
4731:
4576:
4488:
4322:
519:
triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular
5819:
5750:
Schirra, Stefan (2008). "How
Reliable Are Practical Point-in-Polygon Strategies?". In Halperin, Dan; Mehlhorn, Kurt (eds.).
611:
3320:
representing the various elements (sides, vertices, etc.) and their connectivity. A real geometric polygon is said to be a
773:-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple
5807:
5248:
Benjamin, Elliot; Snyder, C. Mathematical
Proceedings of the Cambridge Philosophical Society 156.3 (May 2014): 409–424.;
4705:
526:
5795:
4645:
5976:
5956:
4290:
4171:
2213:
1528:
of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive
5433:
3324:
of the associated abstract polygon. Depending on the mapping, all the generalizations described here can be realized.
2162:
5686:, Castellani Halls, Capitoline Museum, accessed 2013-11-11. Two pentagrams are visible near the center of the image,
5951:
5908:
5883:
5494:
5479:
4362:
1525:
5736:
4370:
4311:
465:
Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are:
4662:
4369:, as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and
3247:
6011:
4600:
4581:
4318:
3567:
3294:
2117:
20:
4687:
B.Sz. Nagy, L. Rédey: Eine
Verallgemeinerung der Inhaltsformel von Heron. Publ. Math. Debrecen 1, 42–50 (1949)
3461:
Not generally recognised as a polygon, although some disciplines such as graph theory sometimes use the term.
654:
5752:
Algorithms - ESA 2008: 16th Annual
European Symposium, Karlsruhe, Germany, September 15-17, 2008, Proceedings
2165:
asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon.
703:
6387:
5936:
4193:
3989:
3043:), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for
5005:{\displaystyle \lim _{n\to +\infty }R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=\pi \cdot R^{2}}
574:
5961:
5846:
5296:
4419:
3788:
The simplest polygon such that it is not known if the regular form can be constructed with neusis or not.
91:
6931:
6924:
6917:
6362:
6302:
5941:
4533:
4498:
4411:
is large, this approaches one half. Or, each vertex inside the square mesh connects four edges (lines).
3941:
3883:
3709:
3563:
3343:
are used in any dimension, with the distinction between the two that a polytope is necessarily bounded.)
3317:
423:
5290:
3626:
The simplest polygon such that the regular form cannot be constructed with compass, straightedge, and
6588:
6535:
6246:
6016:
5946:
5888:
4425:
3238:
is a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allows the
2283:
2122:
617:
2088:{\displaystyle {\begin{aligned}A={\frac {1}{2}}(a_{1}\\{}+a_{2}\\{}+\cdots +a_{n-2}).\end{aligned}}}
1476:
317:: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.
6943:
6842:
6592:
6352:
6327:
6297:
6292:
6251:
5966:
5646:
4358:
4255:
4247:
4235:, where the angles between the sides depend on the type of mineral from which the crystal is made.
4221:
3761:
3571:
1430:
758:
409:
333:
327:
280:
80:
2101:
If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points,
6812:
6762:
6712:
6669:
6639:
6599:
6562:
6380:
6357:
5898:
5722:
5129:
4868:
4842:
4754:
4548:
4186:
3283:
2999:{\displaystyle C_{y}={\frac {1}{6A}}\sum _{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}).}
2827:{\displaystyle C_{x}={\frac {1}{6A}}\sum _{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}),}
790:
454:
359:
3230:
The idea of a polygon has been generalized in various ways. Some of the more important include:
2180:-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).
412:: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
6976:
6951:
6337:
5931:
5839:
5778:
5656:
5650:
5629:
5624:
5606:
5568:
5563:
5545:
5522:
5499:
5461:
5438:
5412:
5391:
5224:
5064:
5056:
5023:
4668:
4629:
4624:
A new universal etymological technological, and pronouncing dictionary of the
English language
4538:
4366:
4354:
4179:
3899:
3493:
3475:
3313:
3251:
3235:
2430:
2102:
782:
370:
109:
5601:
5586:
5540:
5517:
5375:
5218:
1397:
6955:
6520:
6509:
6498:
6487:
6478:
6469:
6456:
6434:
6422:
6408:
6404:
5866:
5755:
5351:
5264:
5119:
4898:
4852:
4746:
4518:
4475:
4251:
4225:
4205:
4201:
3545:
3380:
3306:
3290:
2200:
1542:
983:
415:
363:
299:
47:
5825:
5141:
4864:
6545:
6530:
6332:
6312:
6307:
6277:
5996:
5971:
5903:
5683:
5312:
5137:
4860:
4508:
4155:
3981:
3958:
3627:
3418:
2635:
2189:
993:
766:
381:
339:
274:
176:
99:
3474:
Not generally recognised as a polygon in the
Euclidean plane, although it can exist as a
4889:(R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
4216:
6895:
6342:
6322:
6287:
6282:
5913:
5893:
4641:
4503:
4493:
3985:
3376:
3364:
3276:
3262:
3012:
1579:
1537:
989:
794:
753:
500:
470:
385:
374:
345:
308:
268:
259:
207:
136:
4794:
4474:
lies inside a simple polygon given by a sequence of line segments. This is called the
4361:
used in modelling and rendering. They are defined in a database, containing arrays of
3527:
The simplest polygon which can exist as a regular star. A star pentagon is known as a
806:
6970:
6912:
6800:
6793:
6786:
6750:
6743:
6736:
6700:
6693:
6417:
6317:
6168:
6061:
5981:
5923:
5405:
4558:
3903:
3769:
3511:
3501:
3404:
3298:
304:
5813:
271:: the boundary of the polygon does not cross itself. All convex polygons are simple.
6852:
6347:
6217:
6173:
6137:
6127:
6122:
5268:
4872:
4553:
4543:
4528:
4377:
4163:
3793:
3700:
3674:
3661:
3421:
3258:
607:
435:
396:
353:
314:
180:
168:
87:
5801:
5124:
4762:
5759:
5754:. Lecture Notes in Computer Science. Vol. 5193. Springer. pp. 744–755.
5456:
4622:
3923:
Used as an example in some philosophical discussions, for example in
Descartes's
814:
In this section, the vertices of the polygon under consideration are taken to be
6861:
6822:
6772:
6722:
6679:
6649:
6581:
6567:
6256:
6163:
6142:
6132:
4831:(2005). "The area of cyclic polygons: recent progress on Robbins' conjectures".
4300:
4197:
3751:
3716:
3687:
3302:
3243:
1184:
610:
were first studied by
Poinsot, in the same paper in which he describes the four
277:: Non-convex and simple. There is at least one interior angle greater than 180°.
28:
5249:
4185:
The first known systematic study of non-convex polygons in general was made by
2176:-gons with given side lengths, the one with the largest area is cyclic. Of all
6847:
6831:
6781:
6731:
6688:
6658:
6572:
6261:
6117:
6107:
5991:
5340:"Nominalism and constructivism in seventeenth-century mathematical philosophy"
5024:"Slaying a geometrical 'Monster': finding the area of a crossed Quadrilateral"
4856:
4259:
3907:
3648:
3617:
3357:
3328:
5781:
5356:
5339:
4795:"Dergiades, Nikolaos, "An elementary proof of the isoperimetric inequality",
4783:. translators: J Massalski and C Mills Jr. D C Heath and Company: Boston, MA.
3261:
does not lie in a flat plane, but zigzags in three (or more) dimensions. The
6903:
6817:
6767:
6717:
6674:
6644:
6613:
6236:
6226:
6203:
6193:
6183:
6112:
6021:
5986:
5786:
4270:
4159:
3966:
3933:
3890:
3635:
3528:
3428:
3395:
3279:
is an infinite sequence of sides and angles that do not lie in a flat plane.
3269:
778:
160:. In contexts where one is concerned only with simple and solid polygons, a
3760:
The simplest polygon such that the regular form cannot be constructed with
255:
Polygons may be characterized by their convexity or type of non-convexity:
234:
6877:
6632:
6628:
6555:
6241:
6231:
6188:
6147:
6076:
6066:
6056:
5875:
4828:
4650:. Pure and Applied Mathematics. Vol. 61. Academic Press. p. 37.
4523:
3915:
3739:
3553:
3519:
3483:
3424:
3400:
3389:
3332:
3051:
3033:
187:
129:
37:
4231:
Polygons appear in rock formations, most commonly as the flat facets of
6886:
6856:
6623:
6618:
6609:
6550:
6198:
6178:
6091:
6086:
6081:
6071:
6046:
6001:
5434:
The universal book of mathematics: from Abracadabra to Zeno's paradoxes
5133:
4758:
4513:
4395:
squared triangles since there are two triangles in a square. There are
4325: in this section. Unsourced material may be challenged and removed.
4278:
4266:
4232:
3948:
3875:
3605:
3591:
3579:
3536:
3453:
3408:
2264:
1557:
of a simple polygon can also be computed if the lengths of the sides,
6826:
6776:
6726:
6683:
6653:
6604:
6540:
6006:
5820:
Comparison of the different algorithms for Polygon Boolean operations
4847:
4243:
4167:
935:{\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})}
489:
349:
5098:
Discrete and computational geometry: the Goodman-Pollack Festschrift
4750:
5110:
Hass, Joel; Morgan, Frank (1996). "Geodesic nets on the 2-sphere".
1524:
The signed area depends on the ordering of the vertices and of the
6051:
5831:
4215:
4196:
generalized the idea of polygons to the complex plane, where each
4175:
4145:
3466:
3239:
805:
434:
233:
83:
27:
5816:, solutions to mathematical problems computing 2D and 3D polygons
3492:
The simplest polygon which can exist in the Euclidean plane. Can
6576:
4910:
A regular polygon with an infinite number of sides is a circle:
4239:
2638:
can be defined in two different ways, giving different answers:
5835:
4664:
Beyond measure: a guided tour through nature, myth, and number
4294:
4274:
2195:
The area of a regular polygon is given in terms of the radius
5096:
Grunbaum, B.; "Are your polyhedra the same as my polyhedra",
3600:"Nonagon" mixes Latin with Greek; "enneagon" is pure Greek.
3988:
for clarity of concatenated prefix numbers in the naming of
4595:
Grünbaum, B.; Are your polyhedra the same as my polyhedra?
4422:, it is often necessary to determine whether a given point
3215:{\displaystyle c_{y}={\frac {1}{n}}\sum _{i=0}^{n-1}y_{i}.}
3134:{\displaystyle c_{x}={\frac {1}{n}}\sum _{i=0}^{n-1}x_{i},}
485:
164:
may refer only to a simple polygon or to a solid polygon.
5677:
Cratere with the blinding of Polyphemus and a naval battle
4885:
Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in
4597:
Discrete and comput. geom: the Goodman-Pollack festschrift
247:
Polygons are primarily classified by the number of sides.
68:
56:
5602:
On Understanding Understanding: A Philosophy of Knowledge
5544:, Continuum International Publishing Group, 2010, p. 26,
5223:. Springer Science & Business Media. pp. 88–90.
4281:, and the sides and base of each cell are also polygons.
942:
in order. For convenience in some formulas, the notation
186:
A polygon is a 2-dimensional example of the more general
65:
62:
3910:, have used the chiliagon as an example in discussions.
218:) 'corner' or 'angle'. It has been suggested that γόνυ (
175:. Some sources also consider closed polygonal chains in
97:
The segments of a closed polygonal chain are called its
5822:, compares capabilities, speed and numerical robustness
5737:"direct3d rendering, based on vertices & triangles"
4599:, ed. Aronov et al. Springer (2003) pp. 461–488. (
3562:
The simplest polygon such that the regular form is not
289:: the boundary of the polygon crosses itself. The term
183:), even when the chain does not lie in a single plane.
3297:
analogous to an ordinary polygon, which exists in the
2224:
1271:
708:
659:
622:
585:
542:
5808:
How to draw monochrome orthogonal polygons on screens
5292:
The New Elements of Mathematics: Algebra and Geometry
4916:
4428:
3148:
3067:
3015:
2841:
2669:
2459:
2433:
2295:
2216:
2125:
1612:
1479:
1433:
1400:
1195:
1005:
820:
706:
657:
620:
577:
529:
71:
297:, but this usage risks confusion with the idea of a
107:. The points where two edges meet are the polygon's
59:
53:
6270:
6216:
6156:
6100:
6039:
6030:
5922:
5874:
5814:
comp.graphics.algorithms Frequently Asked Questions
5802:
Polygons, types of polygons, and polygon properties
4585:, Methuen and Co., 1948 (3rd Edition, Dover, 1973).
4166:), appearing as early as the 7th century B.C. on a
50:
5404:
5212:
5210:
5208:
5206:
5204:
5202:
5200:
5198:
5196:
5194:
5192:
5190:
5188:
5186:
5184:
5004:
4466:
4154:Polygons have been known since ancient times. The
3214:
3133:
3021:
2998:
2826:
2618:
2442:
2423:-gon inscribed in a unit-radius circle, with side
2408:
2252:
2150:
2087:
1514:
1465:
1419:
1383:
1172:
988:If the polygon is non-self-intersecting (that is,
934:
741:
692:
635:
598:
564:{\displaystyle \left(1-{\tfrac {2}{n}}\right)\pi }
563:
167:A polygonal chain may cross over itself, creating
5826:Interior angle sum of polygons: a general formula
5605:, 2nd ed, Fordham University Press, 1993, p. 86,
5182:
5180:
5178:
5176:
5174:
5172:
5170:
5168:
5166:
5164:
4376:Any surface is modelled as a tessellation called
3265:of the regular polytopes are well known examples.
388:. The polygon is also equilateral and tangential.
190:in any number of dimensions. There are many more
5411:(Online-Ausg. ed.). New York: McGraw-Hill.
5112:Proceedings of the American Mathematical Society
4918:
4699:"Calculating The Area And Centroid Of A Polygon"
2253:{\displaystyle A={\tfrac {1}{2}}\cdot p\cdot r.}
2188:Many specialized formulas apply to the areas of
426:to L intersects the polygon not more than twice.
4238:Regular hexagons can occur when the cooling of
3974:A degenerate polygon of infinitely many sides.
2161:For any two simple polygons of equal area, the
2098:The formula was described by Lopshits in 1963.
761:to the interior angle. Tracing around a convex
3630:. However, it can be constructed with neusis.
6388:
5847:
5708:Shephard, G.C.; "Regular complex polytopes",
5100:, Ed. Aronov et al., Springer (2003), p. 464.
474:– The sum of the interior angles of a simple
377:. The polygon is also cyclic and equiangular.
8:
5628:, reprint edition, Routledge, 2004, p. 202,
5250:https://dx.doi.org/10.1017/S0305004113000753
5083:
5081:
5079:
5077:
3009:In these formulas, the signed value of area
2450:can also be expressed trigonometrically as:
5655:. Courier Dover Publications. p. 162.
5244:
5242:
5240:
4814:Robbins, "Polygons inscribed in a circle",
4158:were known to the ancient Greeks, with the
511:sides ) can be considered to be made up of
6395:
6381:
6373:
6036:
5854:
5840:
5832:
5492:Merrill, John Calhoun and Odell, S. Jack,
5259:
5257:
5059:(1995). "Lectures on Polytopes". Springer
4647:Noneuclidean tesselations and their groups
3436:Polygon names and miscellaneous properties
5567:, Oxford University Press, 2006, p. 124,
5518:An Introduction to Philosophical Analysis
5355:
5307:
5305:
5286:
5284:
5282:
5280:
5278:
5276:
5123:
4996:
4968:
4949:
4940:
4921:
4915:
4846:
4455:
4442:
4427:
4341:Learn how and when to remove this message
4242:forms areas of tightly packed columns of
3570:. However, it can be constructed using a
3203:
3187:
3176:
3162:
3153:
3147:
3122:
3106:
3095:
3081:
3072:
3066:
3014:
2984:
2968:
2949:
2939:
2917:
2904:
2885:
2874:
2855:
2846:
2840:
2812:
2796:
2777:
2767:
2745:
2732:
2713:
2702:
2683:
2674:
2668:
2595:
2568:
2535:
2517:
2507:
2494:
2476:
2466:
2458:
2432:
2396:
2377:
2356:
2334:
2315:
2306:
2294:
2223:
2215:
2130:
2124:
2057:
2032:
2013:
1998:
1976:
1957:
1932:
1910:
1897:
1878:
1862:
1843:
1830:
1821:
1799:
1780:
1767:
1742:
1720:
1707:
1688:
1672:
1653:
1640:
1623:
1613:
1611:
1500:
1487:
1478:
1454:
1441:
1432:
1405:
1399:
1346:
1322:
1296:
1278:
1266:
1254:
1243:
1227:
1216:
1203:
1194:
1161:
1148:
1139:
1133:
1120:
1111:
1101:
1085:
1066:
1056:
1037:
1026:
1012:
1004:
917:
898:
873:
860:
841:
828:
819:
707:
705:
658:
656:
621:
619:
584:
576:
541:
528:
144:. The interior of a solid polygon is its
5652:A History of Greek Mathematics, Volume 1
5460:, 2nd ed, Addison-Wesley, 1999. p. 505.
5372:Kant's Metaphysics and Theory of Science
4781:Computation of areas of oriented figures
3994:
3433:
693:{\displaystyle {\tfrac {\pi (p-2q)}{p}}}
606:degrees. The interior angles of regular
6960:List of regular polytopes and compounds
5590:, Sadlier and Co., Boston, 1856, p. 27.
5483:, Loyola University Press, 1928, p. 18.
5437:, John Wiley & Sons, 2004. p. 249.
4613:
2286:can be expressed trigonometrically as:
742:{\displaystyle {\tfrac {180(p-2q)}{p}}}
4899:Area of a regular polygon – derivation
4387:points (vertices) per side, there are
2651:it is treated as two simple triangles.
5388:The Philosophical Works of David Hume
5022:De Villiers, Michael (January 2015).
3997:
599:{\displaystyle 180-{\tfrac {360}{n}}}
16:Plane figure bounded by line segments
7:
5699:, 3rd Edn, Dover (pbk), 1973, p. 114
5319:. The Math Forum – Drexel University
4323:adding citations to reliable sources
3371:), noun use of neuter of πολύγωνος (
3054:of the vertex set of a polygon with
810:Coordinates of a non-convex pentagon
191:
4150:Historical image of polygons (1699)
3782:icosipentagon (or icosikaipentagon)
3335:. (In other conventions, the words
342:: both equilateral and equiangular.
336:: all edges are of the same length.
303:as one which exists in the complex
5521:, 4th ed, Routledge, 1997, p. 56,
5390:, Volume 1, Black and Tait, 1826,
4931:
3835:heptacontagon (or hebdomecontagon)
3805:tetracontagon (or tessaracontagon)
373:: all corners lie within the same
14:
5712:Series 3 Volume 2, 1952, pp 82–97
5031:Learning and Teaching Mathematics
4667:. World Scientific. p. 258.
4627:. Oxford University. p. 404.
4391:squared squares in the mesh, or 2
3992:, though not all sources use it.
3729:enneadecagon (or enneakaidecagon)
781:and 0° for an angular "eight" or
293:is sometimes used in contrast to
5457:College Algebra and Trigonometry
4299:
4162:, a non-convex regular polygon (
3855:enneacontagon (or enenecontagon)
3815:pentacontagon (or pentecontagon)
2108:In every polygon with perimeter
1427:is the squared distance between
384:: all sides lie within the same
194:defined for different purposes.
46:
5798:, with Greek Numerical Prefixes
5541:Key Terms in Philosophy of Mind
5374:, Manchester University Press,
5313:"Naming Polygons and Polyhedra"
4834:Advances in Applied Mathematics
4739:The College Mathematics Journal
4467:{\displaystyle P=(x_{0},y_{0})}
4310:needs additional citations for
4200:dimension is accompanied by an
3926:Meditations on First Philosophy
3845:octacontagon (or ogdoëcontagon)
2263:This radius is also termed its
2151:{\displaystyle p^{2}>4\pi A}
1110:
636:{\displaystyle {\tfrac {p}{q}}}
238:Some different types of polygon
5587:Fundamental Philosophy, Vol II
5269:10.1080/00029890.2002.11919848
4925:
4592:, CUP hbk (1997), pbk. (1999).
4489:Boolean operations on polygons
4461:
4435:
4101:heptaconta- (or hebdomeconta-)
4059:tetraconta- (or tessaraconta-)
3825:hexacontagon (or hexecontagon)
2990:
2932:
2929:
2897:
2818:
2760:
2757:
2725:
2075:
2072:
2069:
2050:
2025:
1991:
1988:
1950:
1916:
1890:
1868:
1855:
1836:
1814:
1811:
1760:
1726:
1700:
1678:
1665:
1646:
1633:
1540:. This is commonly called the
1515:{\displaystyle (x_{j},y_{j}).}
1506:
1480:
1460:
1434:
1107:
1049:
929:
891:
879:
853:
847:
821:
729:
714:
680:
665:
362:: all sides are tangent to an
348:: all corners lie on a single
330:: all corner angles are equal.
222:) 'knee' may be the origin of
1:
5723:"opengl vertex specification"
5625:History of Western Philosophy
5564:The Rise of Modern Philosophy
5158:, Dover Edition (1973), p. 4.
5125:10.1090/S0002-9939-96-03492-2
5061:Graduate Texts in Mathematics
4816:American Mathematical Monthly
4732:"The Surveyor's Area Formula"
4407:vertices per triangle. Where
3058:vertices has the coordinates
1466:{\displaystyle (x_{i},y_{i})}
503:. This is because any simple
418:with respect to a given line
5804:, with interactive animation
5760:10.1007/978-3-540-87744-8_62
5220:The Computer Graphics Manual
4129:enneaconta- (or eneneconta-)
4073:pentaconta- (or penteconta-)
2278:-gon in terms of the radius
2267:and is often represented as
757:– The exterior angle is the
214:) 'much', 'many' and γωνία (
4291:Polygon (computer graphics)
4246:, which may be seen at the
4115:octaconta- (or ogdoëconta-)
2172:determine the area. Of all
192:generalizations of polygons
179:to be a type of polygon (a
7000:
6949:
6376:
4697:Bourke, Paul (July 1988).
4288:
4087:hexaconta- (or hexeconta-)
4031:icosi- (icosa- when alone)
3865:hectogon (or hecatontagon)
981:
651:), each interior angle is
647:-gon with central density
251:Convexity and intersection
173:self-intersecting polygons
18:
5495:Philosophy and Journalism
5477:McCormick, John Francis,
5370:Gottfried Martin (1955),
4901:from Math Open Reference.
4857:10.1016/j.aam.2004.08.006
4418:In computer graphics and
4269:, the surface of the wax
4045:triaconta- (or triconta-)
4022:
4013:
4005:
2636:self-intersecting polygon
777:of 360°, e.g. 720° for a
6982:Euclidean plane geometry
5647:Heath, Sir Thomas Little
5498:, Longman, 1983, p. 47,
5403:Gibilisco, Stan (2003).
5357:10.1016/j.hm.2003.09.002
5338:Sepkoski, David (2005).
4365:(the coordinates of the
3568:compass and straightedge
2443:{\displaystyle \alpha ,}
2118:isoperimetric inequality
307:plane consisting of two
21:Polygon (disambiguation)
5710:Proc. London Math. Soc.
5217:Salomon, David (2011).
4380:. If a square mesh has
4194:Geoffrey Colin Shephard
3898:Philosophers including
1420:{\displaystyle Q_{i,j}}
457:is assumed throughout.
431:Properties and formulas
5480:Scholastic Metaphysics
5297:Charles Sanders Peirce
5006:
4779:A.M. Lopshits (1963).
4661:Kappraff, Jay (2002).
4468:
4420:computational geometry
4228:
4151:
3990:quasiregular polyhedra
3248:Wythoff's construction
3216:
3198:
3135:
3117:
3023:
3000:
2896:
2828:
2724:
2620:
2444:
2419:The area of a regular
2410:
2274:The area of a regular
2254:
2163:Bolyai–Gerwien theorem
2152:
2089:
1532:-axis to the positive
1516:
1467:
1421:
1385:
1265:
1238:
1174:
1048:
936:
811:
743:
694:
637:
612:regular star polyhedra
600:
565:
451:
239:
128:sides; for example, a
92:closed polygonal chain
33:
5047:Coxeter (3rd Ed 1973)
5007:
4534:Polygon triangulation
4499:Constructible polygon
4469:
4219:
4189:in the 14th century.
4149:
3942:Constructible polygon
3884:Constructible polygon
3772:(or icosikaitetragon)
3710:Constructible polygon
3427:is also known as the
3318:partially ordered set
3217:
3172:
3136:
3091:
3024:
3001:
2870:
2829:
2698:
2621:
2445:
2411:
2255:
2153:
2090:
1517:
1468:
1422:
1386:
1239:
1212:
1175:
1022:
982:Further information:
937:
809:
744:
695:
638:
601:
566:
438:
322:Equality and symmetry
237:
31:
6087:Nonagon/Enneagon (9)
6017:Tangential trapezoid
5810:, by Herbert Glarner
5599:Potter, Vincent G.,
5407:Geometry demystified
5344:Historia Mathematica
4914:
4818:102, June–July 1995.
4797:Forum Mathematicorum
4730:Bart Braden (1986).
4711:on 16 September 2012
4621:Craig, John (1849).
4426:
4367:geometrical vertices
4319:improve this article
3703:(or heptakaidecagon)
3677:(or pentakaidecagon)
3664:(or tetrakaidecagon)
3246:(map making) and in
3146:
3065:
3013:
2839:
2667:
2457:
2431:
2293:
2284:circumscribed circle
2214:
2123:
1610:
1477:
1431:
1398:
1193:
1003:
818:
704:
655:
618:
575:
527:
90:connected to form a
19:For other uses, see
6944:pentagonal polytope
6843:Uniform 10-polytope
6403:Fundamental convex
6199:Megagon (1,000,000)
5967:Isosceles trapezoid
5796:What Are Polyhedra?
5622:Russell, Bertrand,
5431:Darling, David J.,
3984:, and advocated by
3754:(or icosikaitrigon)
3719:(or octakaidecagon)
3690:(or hexakaidecagon)
3651:(or triskaidecagon)
3572:neusis construction
3438:
3373:polygōnos/polugōnos
3369:polygōnon/polugōnon
2427:and interior angle
974:will also be used.
759:supplementary angle
6813:Uniform 9-polytope
6763:Uniform 8-polytope
6713:Uniform 7-polytope
6670:Uniform 6-polytope
6640:Uniform 5-polytope
6600:Uniform polychoron
6563:Uniform polyhedron
6411:in dimensions 2–10
6169:Icositetragon (24)
5779:Weisstein, Eric W.
5682:2013-11-12 at the
5454:Dugopolski, Mark,
5002:
4935:
4887:Mathematical Plums
4549:Synthetic geometry
4464:
4229:
4187:Thomas Bradwardine
4152:
3434:
3284:polygon with holes
3212:
3131:
3019:
2996:
2824:
2616:
2440:
2406:
2250:
2233:
2203:and its perimeter
2148:
2085:
2083:
1548:surveyor's formula
1512:
1463:
1417:
1381:
1372:
1170:
932:
812:
739:
737:
690:
688:
633:
631:
596:
594:
561:
551:
455:Euclidean geometry
452:
240:
148:, also known as a
124:is a polygon with
34:
6965:
6964:
6952:Polytope families
6409:uniform polytopes
6371:
6370:
6212:
6211:
6189:Myriagon (10,000)
6174:Triacontagon (30)
6138:Heptadecagon (17)
6128:Pentadecagon (15)
6123:Tetradecagon (14)
6062:Quadrilateral (4)
5932:Antiparallelogram
5697:Regular Polytopes
5695:Coxeter, H.S.M.;
5662:978-0-486-24073-2
5418:978-0-07-141650-4
5230:978-0-85729-886-7
5156:Regular polytopes
5154:Coxeter, H.S.M.;
5118:(12): 3843–3850.
5069:978-0-387-94365-7
4981:
4957:
4917:
4799:2, 2002, 129–130"
4674:978-981-02-4702-7
4630:Extract of p. 404
4582:Regular Polytopes
4539:Precision polygon
4357:, a polygon is a
4355:computer graphics
4351:
4350:
4343:
4285:Computer graphics
4180:Capitoline Museum
4139:
4138:
3978:
3977:
3476:spherical polygon
3252:uniform polyhedra
3236:spherical polygon
3170:
3089:
3022:{\displaystyle A}
2868:
2696:
2630:Self-intersecting
2611:
2584:
2551:
2527:
2502:
2486:
2404:
2385:
2347:
2323:
2232:
1631:
1603:are known, from:
1142:
1114:
1020:
783:antiparallelogram
736:
687:
630:
593:
550:
371:vertex-transitive
287:Self-intersecting
210:adjective πολύς (
206:derives from the
6989:
6956:Regular polytope
6517:
6506:
6495:
6454:
6397:
6390:
6383:
6374:
6184:Chiliagon (1000)
6164:Icositrigon (23)
6143:Octadecagon (18)
6133:Hexadecagon (16)
6037:
5856:
5849:
5842:
5833:
5792:
5791:
5764:
5763:
5747:
5741:
5740:
5733:
5727:
5726:
5719:
5713:
5706:
5700:
5693:
5687:
5674:
5668:
5666:
5643:
5637:
5620:
5614:
5597:
5591:
5582:
5576:
5561:Kenny, Anthony,
5559:
5553:
5536:
5530:
5513:
5507:
5490:
5484:
5475:
5469:
5452:
5446:
5429:
5423:
5422:
5410:
5400:
5394:
5384:
5378:
5368:
5362:
5361:
5359:
5335:
5329:
5328:
5326:
5324:
5309:
5300:
5288:
5271:
5261:
5252:
5246:
5235:
5234:
5214:
5159:
5152:
5146:
5145:
5127:
5107:
5101:
5094:
5088:
5085:
5072:
5054:
5048:
5045:
5039:
5038:
5028:
5019:
5013:
5011:
5009:
5008:
5003:
5001:
5000:
4982:
4977:
4969:
4958:
4950:
4945:
4944:
4934:
4908:
4902:
4896:
4890:
4883:
4877:
4876:
4850:
4825:
4819:
4812:
4806:
4805:
4803:
4791:
4785:
4784:
4776:
4770:
4769:
4767:
4761:. Archived from
4736:
4727:
4721:
4720:
4718:
4716:
4710:
4704:. Archived from
4703:
4694:
4688:
4685:
4679:
4678:
4658:
4652:
4651:
4638:
4632:
4628:
4618:
4519:List of polygons
4476:point in polygon
4473:
4471:
4470:
4465:
4460:
4459:
4447:
4446:
4406:
4386:
4346:
4339:
4335:
4332:
4326:
4303:
4295:
4256:Devil's Postpile
4252:Northern Ireland
4248:Giant's Causeway
4226:Northern Ireland
4222:Giant's Causeway
4206:complex polygons
4156:regular polygons
3995:
3959:regular polygons
3439:
3411:are exceptions.
3383:with the suffix
3381:numerical prefix
3316:is an algebraic
3314:abstract polygon
3221:
3219:
3218:
3213:
3208:
3207:
3197:
3186:
3171:
3163:
3158:
3157:
3140:
3138:
3137:
3132:
3127:
3126:
3116:
3105:
3090:
3082:
3077:
3076:
3057:
3049:
3042:
3028:
3026:
3025:
3020:
3005:
3003:
3002:
2997:
2989:
2988:
2979:
2978:
2960:
2959:
2944:
2943:
2928:
2927:
2909:
2908:
2895:
2884:
2869:
2867:
2856:
2851:
2850:
2833:
2831:
2830:
2825:
2817:
2816:
2807:
2806:
2788:
2787:
2772:
2771:
2756:
2755:
2737:
2736:
2723:
2712:
2697:
2695:
2684:
2679:
2678:
2625:
2623:
2622:
2617:
2612:
2610:
2596:
2585:
2583:
2569:
2552:
2550:
2536:
2528:
2523:
2522:
2521:
2508:
2503:
2495:
2487:
2482:
2481:
2480:
2467:
2449:
2447:
2446:
2441:
2415:
2413:
2412:
2407:
2405:
2397:
2386:
2378:
2361:
2360:
2348:
2343:
2335:
2324:
2316:
2311:
2310:
2259:
2257:
2256:
2251:
2234:
2225:
2201:inscribed circle
2190:regular polygons
2184:Regular polygons
2157:
2155:
2154:
2149:
2135:
2134:
2094:
2092:
2091:
2086:
2084:
2068:
2067:
2043:
2042:
2024:
2023:
1999:
1987:
1986:
1962:
1961:
1943:
1942:
1915:
1914:
1902:
1901:
1883:
1882:
1867:
1866:
1848:
1847:
1835:
1834:
1822:
1810:
1809:
1785:
1784:
1772:
1771:
1753:
1752:
1725:
1724:
1712:
1711:
1693:
1692:
1677:
1676:
1658:
1657:
1645:
1644:
1632:
1624:
1543:shoelace formula
1535:
1531:
1521:
1519:
1518:
1513:
1505:
1504:
1492:
1491:
1472:
1470:
1469:
1464:
1459:
1458:
1446:
1445:
1426:
1424:
1423:
1418:
1416:
1415:
1390:
1388:
1387:
1382:
1377:
1376:
1369:
1368:
1339:
1338:
1313:
1312:
1289:
1288:
1264:
1253:
1237:
1226:
1208:
1207:
1179:
1177:
1176:
1171:
1166:
1165:
1153:
1152:
1143:
1140:
1138:
1137:
1125:
1124:
1115:
1112:
1106:
1105:
1096:
1095:
1077:
1076:
1061:
1060:
1047:
1036:
1021:
1013:
984:Shoelace formula
973:
941:
939:
938:
933:
928:
927:
909:
908:
878:
877:
865:
864:
846:
845:
833:
832:
748:
746:
745:
740:
738:
732:
709:
699:
697:
696:
691:
689:
683:
660:
642:
640:
639:
634:
632:
623:
614:: for a regular
605:
603:
602:
597:
595:
586:
570:
568:
567:
562:
557:
553:
552:
543:
518:
499:
488:
449:
439:Partitioning an
364:inscribed circle
151:polygonal region
78:
77:
74:
73:
70:
67:
64:
61:
58:
55:
52:
6999:
6998:
6992:
6991:
6990:
6988:
6987:
6986:
6967:
6966:
6935:
6928:
6921:
6804:
6797:
6790:
6754:
6747:
6740:
6704:
6697:
6531:Regular polygon
6524:
6515:
6508:
6504:
6497:
6493:
6484:
6475:
6468:
6464:
6452:
6446:
6442:
6430:
6412:
6401:
6372:
6367:
6266:
6220:
6208:
6152:
6118:Tridecagon (13)
6108:Hendecagon (11)
6096:
6032:
6026:
5997:Right trapezoid
5918:
5870:
5860:
5777:
5776:
5773:
5768:
5767:
5749:
5748:
5744:
5735:
5734:
5730:
5721:
5720:
5716:
5707:
5703:
5694:
5690:
5684:Wayback Machine
5675:
5671:
5663:
5645:
5644:
5640:
5621:
5617:
5598:
5594:
5584:Balmes, James,
5583:
5579:
5560:
5556:
5537:
5533:
5515:Hospers, John,
5514:
5510:
5491:
5487:
5476:
5472:
5453:
5449:
5430:
5426:
5419:
5402:
5401:
5397:
5385:
5381:
5369:
5365:
5337:
5336:
5332:
5322:
5320:
5311:
5310:
5303:
5289:
5274:
5262:
5255:
5247:
5238:
5231:
5216:
5215:
5162:
5153:
5149:
5109:
5108:
5104:
5095:
5091:
5086:
5075:
5055:
5051:
5046:
5042:
5026:
5021:
5020:
5016:
4992:
4970:
4936:
4912:
4911:
4909:
4905:
4897:
4893:
4884:
4880:
4827:
4826:
4822:
4813:
4809:
4801:
4793:
4792:
4788:
4778:
4777:
4773:
4765:
4751:10.2307/2686282
4734:
4729:
4728:
4724:
4714:
4712:
4708:
4701:
4696:
4695:
4691:
4686:
4682:
4675:
4660:
4659:
4655:
4642:Magnus, Wilhelm
4640:
4639:
4635:
4620:
4619:
4615:
4610:
4577:Coxeter, H.S.M.
4573:
4568:
4563:
4509:Geometric shape
4484:
4451:
4438:
4424:
4423:
4396:
4381:
4347:
4336:
4330:
4327:
4316:
4304:
4293:
4287:
4277:is an array of
4214:
4204:one, to create
4178:and now in the
4144:
3638:(or duodecagon)
3628:angle trisector
3363:(a noun), from
3350:
3291:complex polygon
3263:Petrie polygons
3228:
3226:Generalizations
3199:
3149:
3144:
3143:
3118:
3068:
3063:
3062:
3055:
3044:
3037:
3011:
3010:
2980:
2964:
2945:
2935:
2913:
2900:
2860:
2842:
2837:
2836:
2808:
2792:
2773:
2763:
2741:
2728:
2688:
2670:
2665:
2664:
2658:
2632:
2600:
2573:
2540:
2513:
2509:
2472:
2468:
2455:
2454:
2429:
2428:
2352:
2336:
2302:
2291:
2290:
2212:
2211:
2186:
2126:
2121:
2120:
2082:
2081:
2053:
2028:
2009:
1995:
1994:
1972:
1953:
1928:
1906:
1893:
1874:
1858:
1839:
1826:
1818:
1817:
1795:
1776:
1763:
1738:
1716:
1703:
1684:
1668:
1649:
1636:
1608:
1607:
1601:
1595:
1588:
1580:exterior angles
1576:
1570:
1563:
1533:
1529:
1496:
1483:
1475:
1474:
1450:
1437:
1429:
1428:
1401:
1396:
1395:
1371:
1370:
1342:
1340:
1318:
1315:
1314:
1292:
1290:
1274:
1267:
1199:
1191:
1190:
1157:
1144:
1141: and
1129:
1116:
1097:
1081:
1062:
1052:
1001:
1000:
986:
980:
978:Simple polygons
971:
964:
956:
949:
943:
913:
894:
869:
856:
837:
824:
816:
815:
804:
797:of the polygon.
710:
702:
701:
661:
653:
652:
616:
615:
573:
572:
534:
530:
525:
524:
512:
493:
479:
463:
444:
433:
406:
382:edge-transitive
324:
300:complex polygon
253:
245:
243:Number of sides
232:
200:
177:Euclidean space
49:
45:
24:
17:
12:
11:
5:
6997:
6996:
6993:
6985:
6984:
6979:
6969:
6968:
6963:
6962:
6947:
6946:
6937:
6933:
6926:
6919:
6915:
6906:
6889:
6880:
6869:
6868:
6866:
6864:
6859:
6850:
6845:
6839:
6838:
6836:
6834:
6829:
6820:
6815:
6809:
6808:
6806:
6802:
6795:
6788:
6784:
6779:
6770:
6765:
6759:
6758:
6756:
6752:
6745:
6738:
6734:
6729:
6720:
6715:
6709:
6708:
6706:
6702:
6695:
6691:
6686:
6677:
6672:
6666:
6665:
6663:
6661:
6656:
6647:
6642:
6636:
6635:
6626:
6621:
6616:
6607:
6602:
6596:
6595:
6586:
6584:
6579:
6570:
6565:
6559:
6558:
6553:
6548:
6543:
6538:
6533:
6527:
6526:
6522:
6518:
6513:
6502:
6491:
6482:
6473:
6466:
6460:
6450:
6444:
6438:
6432:
6426:
6420:
6414:
6413:
6402:
6400:
6399:
6392:
6385:
6377:
6369:
6368:
6366:
6365:
6360:
6355:
6350:
6345:
6340:
6335:
6330:
6325:
6323:Pseudotriangle
6320:
6315:
6310:
6305:
6300:
6295:
6290:
6285:
6280:
6274:
6272:
6268:
6267:
6265:
6264:
6259:
6254:
6249:
6244:
6239:
6234:
6229:
6223:
6221:
6214:
6213:
6210:
6209:
6207:
6206:
6201:
6196:
6191:
6186:
6181:
6176:
6171:
6166:
6160:
6158:
6154:
6153:
6151:
6150:
6145:
6140:
6135:
6130:
6125:
6120:
6115:
6113:Dodecagon (12)
6110:
6104:
6102:
6098:
6097:
6095:
6094:
6089:
6084:
6079:
6074:
6069:
6064:
6059:
6054:
6049:
6043:
6041:
6034:
6028:
6027:
6025:
6024:
6019:
6014:
6009:
6004:
5999:
5994:
5989:
5984:
5979:
5974:
5969:
5964:
5959:
5954:
5949:
5944:
5939:
5934:
5928:
5926:
5924:Quadrilaterals
5920:
5919:
5917:
5916:
5911:
5906:
5901:
5896:
5891:
5886:
5880:
5878:
5872:
5871:
5861:
5859:
5858:
5851:
5844:
5836:
5830:
5829:
5823:
5817:
5811:
5805:
5799:
5793:
5772:
5771:External links
5769:
5766:
5765:
5742:
5728:
5714:
5701:
5688:
5669:
5661:
5638:
5615:
5592:
5577:
5554:
5538:Mandik, Pete,
5531:
5508:
5485:
5470:
5447:
5424:
5417:
5395:
5379:
5363:
5330:
5301:
5272:
5253:
5236:
5229:
5160:
5147:
5102:
5089:
5073:
5057:Günter Ziegler
5049:
5040:
5014:
4999:
4995:
4991:
4988:
4985:
4980:
4976:
4973:
4967:
4964:
4961:
4956:
4953:
4948:
4943:
4939:
4933:
4930:
4927:
4924:
4920:
4903:
4891:
4878:
4841:(4): 690–696.
4820:
4807:
4786:
4771:
4768:on 2012-11-07.
4745:(4): 326–337.
4722:
4689:
4680:
4673:
4653:
4633:
4612:
4611:
4609:
4606:
4605:
4604:
4593:
4588:Cromwell, P.;
4586:
4572:
4569:
4567:
4564:
4562:
4561:
4556:
4551:
4546:
4541:
4536:
4531:
4526:
4521:
4516:
4511:
4506:
4504:Cyclic polygon
4501:
4496:
4494:Complete graph
4491:
4485:
4483:
4480:
4463:
4458:
4454:
4450:
4445:
4441:
4437:
4434:
4431:
4349:
4348:
4307:
4305:
4298:
4289:Main article:
4286:
4283:
4213:
4210:
4143:
4140:
4137:
4136:
4133:
4130:
4127:
4123:
4122:
4119:
4116:
4113:
4109:
4108:
4105:
4102:
4099:
4095:
4094:
4091:
4088:
4085:
4081:
4080:
4077:
4074:
4071:
4067:
4066:
4063:
4060:
4057:
4053:
4052:
4049:
4046:
4043:
4039:
4038:
4035:
4032:
4029:
4025:
4024:
4021:
4018:
4015:
4011:
4010:
4007:
4004:
3999:
3986:John H. Conway
3976:
3975:
3972:
3969:
3963:
3962:
3954:
3951:
3945:
3944:
3939:
3936:
3930:
3929:
3921:
3918:
3912:
3911:
3900:René Descartes
3896:
3893:
3887:
3886:
3881:
3878:
3872:
3871:
3869:
3866:
3862:
3861:
3859:
3856:
3852:
3851:
3849:
3846:
3842:
3841:
3839:
3836:
3832:
3831:
3829:
3826:
3822:
3821:
3819:
3816:
3812:
3811:
3809:
3806:
3802:
3801:
3799:
3796:
3790:
3789:
3786:
3783:
3779:
3778:
3776:
3773:
3766:
3765:
3758:
3755:
3748:
3747:
3745:
3742:
3736:
3735:
3733:
3730:
3726:
3725:
3723:
3720:
3713:
3712:
3707:
3704:
3697:
3696:
3694:
3691:
3684:
3683:
3681:
3678:
3671:
3670:
3668:
3665:
3658:
3657:
3655:
3652:
3645:
3644:
3642:
3639:
3632:
3631:
3624:
3621:
3620:(or undecagon)
3614:
3613:
3611:
3608:
3602:
3601:
3598:
3595:
3588:
3587:
3585:
3582:
3576:
3575:
3560:
3557:
3550:
3549:
3542:
3539:
3533:
3532:
3525:
3522:
3516:
3515:
3508:
3505:
3498:
3497:
3490:
3487:
3480:
3479:
3472:
3469:
3463:
3462:
3459:
3456:
3450:
3449:
3446:
3443:
3349:
3346:
3345:
3344:
3325:
3310:
3287:
3280:
3277:skew apeirogon
3273:
3266:
3255:
3227:
3224:
3223:
3222:
3211:
3206:
3202:
3196:
3193:
3190:
3185:
3182:
3179:
3175:
3169:
3166:
3161:
3156:
3152:
3141:
3130:
3125:
3121:
3115:
3112:
3109:
3104:
3101:
3098:
3094:
3088:
3085:
3080:
3075:
3071:
3029:must be used.
3018:
3007:
3006:
2995:
2992:
2987:
2983:
2977:
2974:
2971:
2967:
2963:
2958:
2955:
2952:
2948:
2942:
2938:
2934:
2931:
2926:
2923:
2920:
2916:
2912:
2907:
2903:
2899:
2894:
2891:
2888:
2883:
2880:
2877:
2873:
2866:
2863:
2859:
2854:
2849:
2845:
2834:
2823:
2820:
2815:
2811:
2805:
2802:
2799:
2795:
2791:
2786:
2783:
2780:
2776:
2770:
2766:
2762:
2759:
2754:
2751:
2748:
2744:
2740:
2735:
2731:
2727:
2722:
2719:
2716:
2711:
2708:
2705:
2701:
2694:
2691:
2687:
2682:
2677:
2673:
2657:
2654:
2653:
2652:
2648:
2634:The area of a
2631:
2628:
2627:
2626:
2615:
2609:
2606:
2603:
2599:
2594:
2591:
2588:
2582:
2579:
2576:
2572:
2567:
2564:
2561:
2558:
2555:
2549:
2546:
2543:
2539:
2534:
2531:
2526:
2520:
2516:
2512:
2506:
2501:
2498:
2493:
2490:
2485:
2479:
2475:
2471:
2465:
2462:
2439:
2436:
2417:
2416:
2403:
2400:
2395:
2392:
2389:
2384:
2381:
2376:
2373:
2370:
2367:
2364:
2359:
2355:
2351:
2346:
2342:
2339:
2333:
2330:
2327:
2322:
2319:
2314:
2309:
2305:
2301:
2298:
2261:
2260:
2249:
2246:
2243:
2240:
2237:
2231:
2228:
2222:
2219:
2185:
2182:
2147:
2144:
2141:
2138:
2133:
2129:
2103:Pick's theorem
2096:
2095:
2080:
2077:
2074:
2071:
2066:
2063:
2060:
2056:
2052:
2049:
2046:
2041:
2038:
2035:
2031:
2027:
2022:
2019:
2016:
2012:
2008:
2005:
2002:
1997:
1996:
1993:
1990:
1985:
1982:
1979:
1975:
1971:
1968:
1965:
1960:
1956:
1952:
1949:
1946:
1941:
1938:
1935:
1931:
1927:
1924:
1921:
1918:
1913:
1909:
1905:
1900:
1896:
1892:
1889:
1886:
1881:
1877:
1873:
1870:
1865:
1861:
1857:
1854:
1851:
1846:
1842:
1838:
1833:
1829:
1825:
1820:
1819:
1816:
1813:
1808:
1805:
1802:
1798:
1794:
1791:
1788:
1783:
1779:
1775:
1770:
1766:
1762:
1759:
1756:
1751:
1748:
1745:
1741:
1737:
1734:
1731:
1728:
1723:
1719:
1715:
1710:
1706:
1702:
1699:
1696:
1691:
1687:
1683:
1680:
1675:
1671:
1667:
1664:
1661:
1656:
1652:
1648:
1643:
1639:
1635:
1630:
1627:
1622:
1619:
1616:
1615:
1599:
1593:
1586:
1574:
1568:
1561:
1538:absolute value
1511:
1508:
1503:
1499:
1495:
1490:
1486:
1482:
1462:
1457:
1453:
1449:
1444:
1440:
1436:
1414:
1411:
1408:
1404:
1392:
1391:
1380:
1375:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1345:
1341:
1337:
1334:
1331:
1328:
1325:
1321:
1317:
1316:
1311:
1308:
1305:
1302:
1299:
1295:
1291:
1287:
1284:
1281:
1277:
1273:
1272:
1270:
1263:
1260:
1257:
1252:
1249:
1246:
1242:
1236:
1233:
1230:
1225:
1222:
1219:
1215:
1211:
1206:
1202:
1198:
1181:
1180:
1169:
1164:
1160:
1156:
1151:
1147:
1136:
1132:
1128:
1123:
1119:
1109:
1104:
1100:
1094:
1091:
1088:
1084:
1080:
1075:
1072:
1069:
1065:
1059:
1055:
1051:
1046:
1043:
1040:
1035:
1032:
1029:
1025:
1019:
1016:
1011:
1008:
992:), the signed
979:
976:
969:
962:
954:
947:
931:
926:
923:
920:
916:
912:
907:
904:
901:
897:
893:
890:
887:
884:
881:
876:
872:
868:
863:
859:
855:
852:
849:
844:
840:
836:
831:
827:
823:
803:
800:
799:
798:
795:turning number
754:Exterior angle
750:
735:
731:
728:
725:
722:
719:
716:
713:
686:
682:
679:
676:
673:
670:
667:
664:
629:
626:
592:
589:
583:
580:
560:
556:
549:
546:
540:
537:
533:
507:-gon ( having
471:Interior angle
462:
459:
432:
429:
428:
427:
413:
405:
402:
390:
389:
386:symmetry orbit
378:
375:symmetry orbit
367:
357:
343:
337:
331:
323:
320:
319:
318:
312:
284:
278:
272:
266:
263:
252:
249:
244:
241:
231:
230:Classification
228:
199:
196:
157:polygonal area
137:simple polygon
15:
13:
10:
9:
6:
4:
3:
2:
6995:
6994:
6983:
6980:
6978:
6975:
6974:
6972:
6961:
6957:
6953:
6948:
6945:
6941:
6938:
6936:
6929:
6922:
6916:
6914:
6910:
6907:
6905:
6901:
6897:
6893:
6890:
6888:
6884:
6881:
6879:
6875:
6871:
6870:
6867:
6865:
6863:
6860:
6858:
6854:
6851:
6849:
6846:
6844:
6841:
6840:
6837:
6835:
6833:
6830:
6828:
6824:
6821:
6819:
6816:
6814:
6811:
6810:
6807:
6805:
6798:
6791:
6785:
6783:
6780:
6778:
6774:
6771:
6769:
6766:
6764:
6761:
6760:
6757:
6755:
6748:
6741:
6735:
6733:
6730:
6728:
6724:
6721:
6719:
6716:
6714:
6711:
6710:
6707:
6705:
6698:
6692:
6690:
6687:
6685:
6681:
6678:
6676:
6673:
6671:
6668:
6667:
6664:
6662:
6660:
6657:
6655:
6651:
6648:
6646:
6643:
6641:
6638:
6637:
6634:
6630:
6627:
6625:
6622:
6620:
6619:Demitesseract
6617:
6615:
6611:
6608:
6606:
6603:
6601:
6598:
6597:
6594:
6590:
6587:
6585:
6583:
6580:
6578:
6574:
6571:
6569:
6566:
6564:
6561:
6560:
6557:
6554:
6552:
6549:
6547:
6544:
6542:
6539:
6537:
6534:
6532:
6529:
6528:
6525:
6519:
6516:
6512:
6505:
6501:
6494:
6490:
6485:
6481:
6476:
6472:
6467:
6465:
6463:
6459:
6449:
6445:
6443:
6441:
6437:
6433:
6431:
6429:
6425:
6421:
6419:
6416:
6415:
6410:
6406:
6398:
6393:
6391:
6386:
6384:
6379:
6378:
6375:
6364:
6363:Weakly simple
6361:
6359:
6356:
6354:
6351:
6349:
6346:
6344:
6341:
6339:
6336:
6334:
6331:
6329:
6326:
6324:
6321:
6319:
6316:
6314:
6311:
6309:
6306:
6304:
6303:Infinite skew
6301:
6299:
6296:
6294:
6291:
6289:
6286:
6284:
6281:
6279:
6276:
6275:
6273:
6269:
6263:
6260:
6258:
6255:
6253:
6250:
6248:
6245:
6243:
6240:
6238:
6235:
6233:
6230:
6228:
6225:
6224:
6222:
6219:
6218:Star polygons
6215:
6205:
6204:Apeirogon (∞)
6202:
6200:
6197:
6195:
6192:
6190:
6187:
6185:
6182:
6180:
6177:
6175:
6172:
6170:
6167:
6165:
6162:
6161:
6159:
6155:
6149:
6148:Icosagon (20)
6146:
6144:
6141:
6139:
6136:
6134:
6131:
6129:
6126:
6124:
6121:
6119:
6116:
6114:
6111:
6109:
6106:
6105:
6103:
6099:
6093:
6090:
6088:
6085:
6083:
6080:
6078:
6075:
6073:
6070:
6068:
6065:
6063:
6060:
6058:
6055:
6053:
6050:
6048:
6045:
6044:
6042:
6038:
6035:
6029:
6023:
6020:
6018:
6015:
6013:
6010:
6008:
6005:
6003:
6000:
5998:
5995:
5993:
5990:
5988:
5985:
5983:
5982:Parallelogram
5980:
5978:
5977:Orthodiagonal
5975:
5973:
5970:
5968:
5965:
5963:
5960:
5958:
5957:Ex-tangential
5955:
5953:
5950:
5948:
5945:
5943:
5940:
5938:
5935:
5933:
5930:
5929:
5927:
5925:
5921:
5915:
5912:
5910:
5907:
5905:
5902:
5900:
5897:
5895:
5892:
5890:
5887:
5885:
5882:
5881:
5879:
5877:
5873:
5868:
5864:
5857:
5852:
5850:
5845:
5843:
5838:
5837:
5834:
5827:
5824:
5821:
5818:
5815:
5812:
5809:
5806:
5803:
5800:
5797:
5794:
5789:
5788:
5783:
5780:
5775:
5774:
5770:
5761:
5757:
5753:
5746:
5743:
5738:
5732:
5729:
5724:
5718:
5715:
5711:
5705:
5702:
5698:
5692:
5689:
5685:
5681:
5678:
5673:
5670:
5664:
5658:
5654:
5653:
5648:
5642:
5639:
5635:
5634:0-415-32505-6
5631:
5627:
5626:
5619:
5616:
5612:
5611:0-8232-1486-9
5608:
5604:
5603:
5596:
5593:
5589:
5588:
5581:
5578:
5574:
5573:0-19-875277-6
5570:
5566:
5565:
5558:
5555:
5551:
5550:1-84706-349-7
5547:
5543:
5542:
5535:
5532:
5528:
5527:0-415-15792-7
5524:
5520:
5519:
5512:
5509:
5505:
5504:0-582-28157-1
5501:
5497:
5496:
5489:
5486:
5482:
5481:
5474:
5471:
5467:
5466:0-201-34712-1
5463:
5459:
5458:
5451:
5448:
5444:
5443:0-471-27047-4
5440:
5436:
5435:
5428:
5425:
5420:
5414:
5409:
5408:
5399:
5396:
5393:
5389:
5383:
5380:
5377:
5373:
5367:
5364:
5358:
5353:
5349:
5345:
5341:
5334:
5331:
5318:
5314:
5308:
5306:
5302:
5299:(1976), p.298
5298:
5294:
5293:
5287:
5285:
5283:
5281:
5279:
5277:
5273:
5270:
5266:
5260:
5258:
5254:
5251:
5245:
5243:
5241:
5237:
5232:
5226:
5222:
5221:
5213:
5211:
5209:
5207:
5205:
5203:
5201:
5199:
5197:
5195:
5193:
5191:
5189:
5187:
5185:
5183:
5181:
5179:
5177:
5175:
5173:
5171:
5169:
5167:
5165:
5161:
5157:
5151:
5148:
5143:
5139:
5135:
5131:
5126:
5121:
5117:
5113:
5106:
5103:
5099:
5093:
5090:
5084:
5082:
5080:
5078:
5074:
5070:
5066:
5062:
5058:
5053:
5050:
5044:
5041:
5036:
5032:
5025:
5018:
5015:
4997:
4993:
4989:
4986:
4983:
4978:
4974:
4971:
4965:
4962:
4959:
4954:
4951:
4946:
4941:
4937:
4928:
4922:
4907:
4904:
4900:
4895:
4892:
4888:
4882:
4879:
4874:
4870:
4866:
4862:
4858:
4854:
4849:
4844:
4840:
4836:
4835:
4830:
4824:
4821:
4817:
4811:
4808:
4800:
4798:
4790:
4787:
4782:
4775:
4772:
4764:
4760:
4756:
4752:
4748:
4744:
4740:
4733:
4726:
4723:
4707:
4700:
4693:
4690:
4684:
4681:
4676:
4670:
4666:
4665:
4657:
4654:
4649:
4648:
4643:
4637:
4634:
4631:
4626:
4625:
4617:
4614:
4607:
4602:
4598:
4594:
4591:
4587:
4584:
4583:
4578:
4575:
4574:
4570:
4565:
4560:
4559:Tiling puzzle
4557:
4555:
4552:
4550:
4547:
4545:
4542:
4540:
4537:
4535:
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4486:
4481:
4479:
4477:
4456:
4452:
4448:
4443:
4439:
4432:
4429:
4421:
4416:
4412:
4410:
4404:
4400:
4394:
4390:
4384:
4379:
4374:
4372:
4368:
4364:
4360:
4356:
4345:
4342:
4334:
4324:
4320:
4314:
4313:
4308:This section
4306:
4302:
4297:
4296:
4292:
4284:
4282:
4280:
4276:
4272:
4268:
4263:
4261:
4257:
4253:
4249:
4245:
4241:
4236:
4234:
4227:
4223:
4218:
4211:
4209:
4207:
4203:
4199:
4195:
4190:
4188:
4183:
4181:
4177:
4173:
4169:
4165:
4161:
4157:
4148:
4141:
4134:
4131:
4128:
4125:
4124:
4120:
4117:
4114:
4111:
4110:
4106:
4103:
4100:
4097:
4096:
4092:
4089:
4086:
4083:
4082:
4078:
4075:
4072:
4069:
4068:
4064:
4061:
4058:
4055:
4054:
4050:
4047:
4044:
4041:
4040:
4036:
4033:
4030:
4027:
4026:
4019:
4016:
4012:
4009:final suffix
4008:
4003:
4000:
3996:
3993:
3991:
3987:
3983:
3973:
3970:
3968:
3965:
3964:
3961:to a circle.
3960:
3955:
3952:
3950:
3947:
3946:
3943:
3940:
3937:
3935:
3932:
3931:
3928:
3927:
3922:
3919:
3917:
3914:
3913:
3909:
3905:
3904:Immanuel Kant
3901:
3897:
3894:
3892:
3889:
3888:
3885:
3882:
3879:
3877:
3874:
3873:
3870:
3867:
3864:
3863:
3860:
3857:
3854:
3853:
3850:
3847:
3844:
3843:
3840:
3837:
3834:
3833:
3830:
3827:
3824:
3823:
3820:
3817:
3814:
3813:
3810:
3807:
3804:
3803:
3800:
3797:
3795:
3792:
3791:
3787:
3784:
3781:
3780:
3777:
3774:
3771:
3770:icositetragon
3768:
3767:
3763:
3759:
3756:
3753:
3750:
3749:
3746:
3743:
3741:
3738:
3737:
3734:
3731:
3728:
3727:
3724:
3721:
3718:
3715:
3714:
3711:
3708:
3705:
3702:
3699:
3698:
3695:
3692:
3689:
3686:
3685:
3682:
3679:
3676:
3673:
3672:
3669:
3666:
3663:
3660:
3659:
3656:
3653:
3650:
3647:
3646:
3643:
3640:
3637:
3634:
3633:
3629:
3625:
3622:
3619:
3616:
3615:
3612:
3609:
3607:
3604:
3603:
3599:
3596:
3594:(or enneagon)
3593:
3590:
3589:
3586:
3583:
3581:
3578:
3577:
3573:
3569:
3565:
3564:constructible
3561:
3558:
3556:(or septagon)
3555:
3552:
3551:
3547:
3543:
3540:
3538:
3535:
3534:
3531:or pentacle.
3530:
3526:
3523:
3521:
3518:
3517:
3513:
3509:
3506:
3504:(or tetragon)
3503:
3502:quadrilateral
3500:
3499:
3495:
3491:
3488:
3485:
3482:
3481:
3477:
3473:
3470:
3468:
3465:
3464:
3460:
3457:
3455:
3452:
3451:
3447:
3444:
3441:
3440:
3437:
3432:
3430:
3426:
3423:
3420:
3415:
3412:
3410:
3406:
3405:quadrilateral
3402:
3398:
3397:
3392:
3391:
3386:
3382:
3378:
3374:
3370:
3366:
3362:
3359:
3355:
3347:
3342:
3338:
3334:
3330:
3326:
3323:
3319:
3315:
3311:
3308:
3304:
3300:
3299:complex plane
3296:
3295:configuration
3292:
3288:
3285:
3281:
3278:
3274:
3271:
3267:
3264:
3260:
3256:
3253:
3249:
3245:
3241:
3237:
3233:
3232:
3231:
3225:
3209:
3204:
3200:
3194:
3191:
3188:
3183:
3180:
3177:
3173:
3167:
3164:
3159:
3154:
3150:
3142:
3128:
3123:
3119:
3113:
3110:
3107:
3102:
3099:
3096:
3092:
3086:
3083:
3078:
3073:
3069:
3061:
3060:
3059:
3053:
3047:
3040:
3035:
3030:
3016:
2993:
2985:
2981:
2975:
2972:
2969:
2965:
2961:
2956:
2953:
2950:
2946:
2940:
2936:
2924:
2921:
2918:
2914:
2910:
2905:
2901:
2892:
2889:
2886:
2881:
2878:
2875:
2871:
2864:
2861:
2857:
2852:
2847:
2843:
2835:
2821:
2813:
2809:
2803:
2800:
2797:
2793:
2789:
2784:
2781:
2778:
2774:
2768:
2764:
2752:
2749:
2746:
2742:
2738:
2733:
2729:
2720:
2717:
2714:
2709:
2706:
2703:
2699:
2692:
2689:
2685:
2680:
2675:
2671:
2663:
2662:
2661:
2655:
2649:
2645:
2641:
2640:
2639:
2637:
2629:
2613:
2607:
2604:
2601:
2597:
2592:
2589:
2586:
2580:
2577:
2574:
2570:
2565:
2562:
2559:
2556:
2553:
2547:
2544:
2541:
2537:
2532:
2529:
2524:
2518:
2514:
2510:
2504:
2499:
2496:
2491:
2488:
2483:
2477:
2473:
2469:
2463:
2460:
2453:
2452:
2451:
2437:
2434:
2426:
2422:
2401:
2398:
2393:
2390:
2387:
2382:
2379:
2374:
2371:
2368:
2365:
2362:
2357:
2353:
2349:
2344:
2340:
2337:
2331:
2328:
2325:
2320:
2317:
2312:
2307:
2303:
2299:
2296:
2289:
2288:
2287:
2285:
2281:
2277:
2272:
2270:
2266:
2247:
2244:
2241:
2238:
2235:
2229:
2226:
2220:
2217:
2210:
2209:
2208:
2206:
2202:
2198:
2193:
2191:
2183:
2181:
2179:
2175:
2171:
2166:
2164:
2159:
2145:
2142:
2139:
2136:
2131:
2127:
2119:
2115:
2111:
2106:
2104:
2099:
2078:
2064:
2061:
2058:
2054:
2047:
2044:
2039:
2036:
2033:
2029:
2020:
2017:
2014:
2010:
2006:
2003:
2000:
1983:
1980:
1977:
1973:
1969:
1966:
1963:
1958:
1954:
1947:
1944:
1939:
1936:
1933:
1929:
1925:
1922:
1919:
1911:
1907:
1903:
1898:
1894:
1887:
1884:
1879:
1875:
1871:
1863:
1859:
1852:
1849:
1844:
1840:
1831:
1827:
1823:
1806:
1803:
1800:
1796:
1792:
1789:
1786:
1781:
1777:
1773:
1768:
1764:
1757:
1754:
1749:
1746:
1743:
1739:
1735:
1732:
1729:
1721:
1717:
1713:
1708:
1704:
1697:
1694:
1689:
1685:
1681:
1673:
1669:
1662:
1659:
1654:
1650:
1641:
1637:
1628:
1625:
1620:
1617:
1606:
1605:
1604:
1602:
1592:
1585:
1581:
1577:
1567:
1560:
1556:
1551:
1549:
1545:
1544:
1539:
1527:
1522:
1509:
1501:
1497:
1493:
1488:
1484:
1455:
1451:
1447:
1442:
1438:
1412:
1409:
1406:
1402:
1378:
1373:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1343:
1335:
1332:
1329:
1326:
1323:
1319:
1309:
1306:
1303:
1300:
1297:
1293:
1285:
1282:
1279:
1275:
1268:
1261:
1258:
1255:
1250:
1247:
1244:
1240:
1234:
1231:
1228:
1223:
1220:
1217:
1213:
1209:
1204:
1200:
1196:
1189:
1188:
1187:
1186:
1167:
1162:
1158:
1154:
1149:
1145:
1134:
1130:
1126:
1121:
1117:
1102:
1098:
1092:
1089:
1086:
1082:
1078:
1073:
1070:
1067:
1063:
1057:
1053:
1044:
1041:
1038:
1033:
1030:
1027:
1023:
1017:
1014:
1009:
1006:
999:
998:
997:
995:
991:
985:
977:
975:
968:
961:
957:
950:
924:
921:
918:
914:
910:
905:
902:
899:
895:
888:
885:
882:
874:
870:
866:
861:
857:
850:
842:
838:
834:
829:
825:
808:
801:
796:
792:
788:
784:
780:
776:
772:
768:
764:
760:
756:
755:
751:
733:
726:
723:
720:
717:
711:
684:
677:
674:
671:
668:
662:
650:
646:
627:
624:
613:
609:
608:star polygons
590:
587:
581:
578:
558:
554:
547:
544:
538:
535:
531:
522:
516:
510:
506:
502:
497:
491:
487:
483:
477:
473:
472:
468:
467:
466:
460:
458:
456:
447:
442:
437:
430:
425:
422:: every line
421:
417:
414:
411:
408:
407:
404:Miscellaneous
403:
401:
399:
398:
387:
383:
379:
376:
372:
368:
365:
361:
358:
355:
352:, called the
351:
347:
344:
341:
338:
335:
332:
329:
326:
325:
321:
316:
313:
310:
306:
302:
301:
296:
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288:
285:
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279:
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273:
270:
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229:
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225:
221:
217:
213:
209:
205:
197:
195:
193:
189:
184:
182:
178:
174:
170:
169:star polygons
165:
163:
159:
158:
153:
152:
147:
143:
142:solid polygon
138:
133:
131:
127:
123:
121:
116:
112:
111:
106:
102:
101:
95:
93:
89:
88:line segments
85:
82:
76:
43:
39:
30:
26:
22:
6939:
6908:
6899:
6891:
6882:
6873:
6853:10-orthoplex
6589:Dodecahedron
6510:
6499:
6488:
6479:
6470:
6461:
6457:
6447:
6439:
6435:
6427:
6423:
6157:>20 sides
6092:Decagon (10)
6077:Heptagon (7)
6067:Pentagon (5)
6057:Triangle (3)
5952:Equidiagonal
5862:
5785:
5751:
5745:
5731:
5717:
5709:
5704:
5696:
5691:
5672:
5651:
5641:
5623:
5618:
5600:
5595:
5585:
5580:
5562:
5557:
5539:
5534:
5516:
5511:
5493:
5488:
5478:
5473:
5455:
5450:
5432:
5427:
5406:
5398:
5387:
5386:David Hume,
5382:
5371:
5366:
5347:
5343:
5333:
5321:. Retrieved
5317:Ask Dr. Math
5316:
5291:
5219:
5155:
5150:
5115:
5111:
5105:
5097:
5092:
5060:
5052:
5043:
5037:(18): 23–28.
5034:
5030:
5017:
4906:
4894:
4886:
4881:
4848:math/0408104
4838:
4832:
4823:
4815:
4810:
4796:
4789:
4780:
4774:
4763:the original
4742:
4738:
4725:
4713:. Retrieved
4706:the original
4692:
4683:
4663:
4656:
4646:
4636:
4623:
4616:
4596:
4589:
4580:
4571:Bibliography
4544:Spirolateral
4529:Polygon soup
4417:
4413:
4408:
4402:
4398:
4392:
4388:
4382:
4378:polygon mesh
4375:
4352:
4337:
4331:October 2018
4328:
4317:Please help
4312:verification
4309:
4264:
4254:, or at the
4237:
4230:
4191:
4184:
4172:Aristophanes
4164:star polygon
4153:
4001:
3979:
3924:
3794:triacontagon
3701:heptadecagon
3675:pentadecagon
3662:tetradecagon
3435:
3416:
3413:
3394:
3388:
3384:
3372:
3368:
3360:
3353:
3351:
3340:
3336:
3321:
3259:skew polygon
3229:
3045:
3038:
3031:
3008:
2659:
2643:
2633:
2424:
2420:
2418:
2279:
2275:
2273:
2268:
2262:
2204:
2196:
2194:
2187:
2177:
2173:
2169:
2167:
2160:
2113:
2109:
2107:
2100:
2097:
1597:
1590:
1583:
1572:
1565:
1558:
1554:
1552:
1547:
1541:
1523:
1393:
1185:determinants
1182:
987:
966:
959:
952:
945:
813:
786:
774:
770:
762:
752:
648:
644:
520:
514:
508:
504:
495:
481:
475:
469:
464:
453:
445:
440:
419:
397:star polygon
394:
391:
380:Isotoxal or
369:Isogonal or
354:circumcircle
315:Star polygon
298:
294:
290:
254:
246:
223:
219:
215:
211:
203:
201:
185:
181:skew polygon
166:
161:
156:
155:
150:
149:
145:
141:
134:
132:is a 3-gon.
125:
119:
118:
114:
108:
104:
98:
96:
41:
35:
25:
6862:10-demicube
6823:9-orthoplex
6773:8-orthoplex
6723:7-orthoplex
6680:6-orthoplex
6650:5-orthoplex
6605:Pentachoron
6593:Icosahedron
6568:Tetrahedron
6353:Star-shaped
6328:Rectilinear
6298:Equilateral
6293:Equiangular
6257:Hendecagram
6101:11–20 sides
6082:Octagon (8)
6072:Hexagon (6)
6047:Monogon (1)
5889:Equilateral
4174:, found at
3752:icositrigon
3717:octadecagon
3688:hexadecagon
3548:the plane.
3514:the plane.
3496:the plane.
3486:(or trigon)
3448:Properties
3367:πολύγωνον (
3356:comes from
3322:realization
3309:dimensions.
3244:cartography
1526:orientation
1113:where
700:radians or
571:radians or
410:Rectilinear
334:Equilateral
328:Equiangular
311:dimensions.
281:Star-shaped
86:made up of
6971:Categories
6848:10-simplex
6832:9-demicube
6782:8-demicube
6732:7-demicube
6689:6-demicube
6659:5-demicube
6573:Octahedron
6358:Tangential
6262:Dodecagram
6040:1–10 sides
6031:By number
6012:Tangential
5992:Right kite
4566:References
4260:California
3908:David Hume
3649:tridecagon
3618:hendecagon
3358:Late Latin
3337:polyhedron
3329:polyhedron
1183:or, using
498:− 2) × 180
443:-gon into
424:orthogonal
360:Tangential
171:and other
6896:orthoplex
6818:9-simplex
6768:8-simplex
6718:7-simplex
6675:6-simplex
6645:5-simplex
6614:Tesseract
6338:Reinhardt
6247:Enneagram
6237:Heptagram
6227:Pentagram
6194:65537-gon
6052:Digon (2)
6022:Trapezoid
5987:Rectangle
5937:Bicentric
5899:Isosceles
5876:Triangles
5787:MathWorld
5782:"Polygon"
5350:: 33–59.
5087:Mathworld
4990:⋅
4987:π
4975:π
4966:
4960:⋅
4947:⋅
4932:∞
4926:→
4829:Pak, Igor
4590:Polyhedra
4401:+ 1) / 2(
4371:materials
4359:primitive
4271:honeycomb
4212:In nature
4202:imaginary
4192:In 1952,
4160:pentagram
3967:apeirogon
3953:1,000,000
3934:65537-gon
3891:chiliagon
3636:dodecagon
3529:pentagram
3429:pentagram
3396:dodecagon
3379:-derived
3361:polygōnum
3352:The word
3333:polytopes
3307:imaginary
3270:apeirogon
3192:−
3174:∑
3111:−
3093:∑
3034:triangles
2962:−
2890:−
2872:∑
2790:−
2718:−
2700:∑
2605:−
2598:α
2593:
2587:⋅
2578:−
2571:α
2566:
2560:⋅
2545:−
2538:α
2533:
2497:π
2492:
2435:α
2399:π
2394:
2388:⋅
2380:π
2375:
2369:⋅
2363:⋅
2341:π
2332:
2326:⋅
2313:⋅
2242:⋅
2236:⋅
2143:π
2112:and area
2062:−
2055:θ
2048:
2037:−
2018:−
2004:⋯
1981:−
1974:θ
1967:⋯
1955:θ
1948:
1937:−
1923:⋯
1908:θ
1895:θ
1888:
1860:θ
1853:
1804:−
1797:θ
1790:⋯
1778:θ
1765:θ
1758:
1747:−
1733:⋯
1718:θ
1705:θ
1698:
1670:θ
1663:
1553:The area
1259:−
1241:∑
1232:−
1214:∑
1079:−
1042:−
1024:∑
922:−
903:−
886:…
779:pentagram
721:−
672:−
663:π
582:−
559:π
539:−
450:triangles
202:The word
198:Etymology
6977:Polygons
6950:Topics:
6913:demicube
6878:polytope
6872:Uniform
6633:600-cell
6629:120-cell
6582:Demicube
6556:Pentagon
6536:Triangle
6313:Isotoxal
6308:Isogonal
6252:Decagram
6242:Octagram
6232:Hexagram
6033:of sides
5962:Harmonic
5863:Polygons
5680:Archived
5649:(1981).
4644:(1974).
4524:Polyform
4482:See also
4363:vertices
4279:hexagons
4273:made by
4233:crystals
4135:-ennea-
4107:-hepta-
4079:-penta-
4065:-tetra-
3916:myriagon
3740:icosagon
3554:heptagon
3520:pentagon
3484:triangle
3425:pentagon
3401:triangle
3390:pentagon
3341:polytope
3305:and two
3052:centroid
2656:Centroid
1578:and the
785:, where
749:degrees.
643:-gon (a
523:-gon is
478:-gon is
416:Monotone
395:regular
188:polytope
130:triangle
110:vertices
38:geometry
6887:simplex
6857:10-cube
6624:24-cell
6610:16-cell
6551:Hexagon
6405:regular
6333:Regular
6278:Concave
6271:Classes
6179:257-gon
6002:Rhombus
5942:Crossed
5392:p. 101.
5142:1343696
5134:2161556
5071:. p. 4.
4873:6756387
4865:2128993
4759:2686282
4514:Golygon
4267:biology
4142:History
4121:-octa-
4093:-hexa-
4020:-hena-
3949:megagon
3876:257-gon
3606:decagon
3592:nonagon
3580:octagon
3537:hexagon
3454:monogon
3419:regular
3409:nonagon
3387:, e.g.
3354:polygon
3301:of two
3250:of the
2647:figure.
2644:density
2282:of its
2265:apothem
2199:of its
2158:holds.
1596:, ...,
1571:, ...,
791:density
789:is the
501:degrees
490:radians
484:− 2) ×
340:Regular
309:complex
305:Hilbert
291:complex
275:Concave
204:polygon
162:polygon
115:corners
79:) is a
42:polygon
6827:9-cube
6777:8-cube
6727:7-cube
6684:6-cube
6654:5-cube
6541:Square
6418:Family
6343:Simple
6288:Cyclic
6283:Convex
6007:Square
5947:Cyclic
5909:Obtuse
5904:Kepler
5659:
5632:
5609:
5571:
5548:
5525:
5502:
5464:
5441:
5415:
5376:p. 22.
5227:
5140:
5132:
5067:
4871:
4863:
4757:
4671:
4554:Tiling
4478:test.
4244:basalt
4168:krater
4051:-tri-
4014:-kai-
3982:Kepler
3938:65,537
3920:10,000
3762:neusis
3445:Sides
3399:. The
3348:Naming
3050:. The
3048:> 3
2116:, the
1394:where
990:simple
461:Angles
350:circle
346:Cyclic
295:simple
269:Simple
260:Convex
84:figure
6546:p-gon
6318:Magic
5914:Right
5894:Ideal
5884:Acute
5323:3 May
5130:JSTOR
5027:(PDF)
4869:S2CID
4843:arXiv
4802:(PDF)
4766:(PDF)
4755:JSTOR
4735:(PDF)
4715:6 Feb
4709:(PDF)
4702:(PDF)
4608:Notes
4224:, in
4176:Caere
4037:-di-
4023:-gon
4006:Ones
3998:Tens
3566:with
3467:digon
3442:Name
3377:Greek
3365:Greek
3293:is a
3240:digon
958:) = (
216:gōnía
212:polús
208:Greek
117:. An
105:sides
100:edges
81:plane
6904:cube
6577:Cube
6407:and
6348:Skew
5972:Kite
5867:List
5657:ISBN
5630:ISBN
5607:ISBN
5569:ISBN
5546:ISBN
5523:ISBN
5500:ISBN
5462:ISBN
5439:ISBN
5413:ISBN
5325:2015
5225:ISBN
5065:ISBN
5035:2015
4717:2013
4669:ISBN
4275:bees
4240:lava
4220:The
4198:real
3895:1000
3546:tile
3544:Can
3512:tile
3494:tile
3422:star
3407:and
3385:-gon
3339:and
3303:real
3032:For
2207:by
2137:>
1473:and
994:area
802:Area
767:turn
517:− 2)
220:gónu
146:body
122:-gon
40:, a
6453:(p)
5756:doi
5352:doi
5295:by
5265:doi
5120:doi
5116:124
4963:sin
4919:lim
4853:doi
4747:doi
4601:pdf
4385:+ 1
4353:In
4321:by
4265:In
4258:in
4250:in
4170:by
4002:and
3880:257
3868:100
3312:An
3268:An
3041:= 3
2590:cos
2563:sin
2530:cot
2489:cot
2391:cos
2372:sin
2329:sin
2045:sin
1945:sin
1885:sin
1850:sin
1755:sin
1695:sin
1660:sin
1546:or
996:is
793:or
712:180
588:360
579:180
492:or
448:− 2
224:gon
154:or
113:or
103:or
36:In
6973::
6958:•
6954:•
6934:21
6930:•
6927:k1
6923:•
6920:k2
6898:•
6855:•
6825:•
6803:21
6799:•
6796:41
6792:•
6789:42
6775:•
6753:21
6749:•
6746:31
6742:•
6739:32
6725:•
6703:21
6699:•
6696:22
6682:•
6652:•
6631:•
6612:•
6591:•
6575:•
6507:/
6496:/
6486:/
6477:/
6455:/
5784:.
5348:32
5346:.
5342:.
5315:.
5304:^
5275:^
5256:^
5239:^
5163:^
5138:MR
5136:.
5128:.
5114:.
5076:^
5063:,
5033:.
5029:.
4867:.
4861:MR
4859:.
4851:.
4839:34
4837:.
4753:.
4743:17
4741:.
4737:.
4579:;
4373:.
4262:.
4208:.
4182:.
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4028:20
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3858:90
3848:80
3838:70
3828:60
3818:50
3808:40
3798:30
3785:25
3775:24
3764:.
3757:23
3744:20
3732:19
3722:18
3706:17
3693:16
3680:15
3667:14
3654:13
3641:12
3623:11
3610:10
3574:.
3478:.
3431:.
3403:,
3393:,
3327:A
3289:A
3282:A
3275:A
3257:A
3234:A
2271:.
2192:.
2170:do
2114:A
1589:,
1582:,
1564:,
1550:.
1197:16
965:,
951:,
400:.
226:.
135:A
94:.
6942:-
6940:n
6932:k
6925:2
6918:1
6911:-
6909:n
6902:-
6900:n
6894:-
6892:n
6885:-
6883:n
6876:-
6874:n
6801:4
6794:2
6787:1
6751:3
6744:2
6737:1
6701:2
6694:1
6523:n
6521:H
6514:2
6511:G
6503:4
6500:F
6492:8
6489:E
6483:7
6480:E
6474:6
6471:E
6462:n
6458:D
6451:2
6448:I
6440:n
6436:B
6428:n
6424:A
6396:e
6389:t
6382:v
5869:)
5865:(
5855:e
5848:t
5841:v
5790:.
5762:.
5758::
5739:.
5725:.
5665:.
5636:.
5613:.
5575:.
5552:.
5529:.
5506:.
5468:.
5445:.
5421:.
5360:.
5354::
5327:.
5267::
5233:.
5144:.
5122::
5012:.
4998:2
4994:R
4984:=
4979:n
4972:2
4955:2
4952:n
4942:2
4938:R
4929:+
4923:n
4875:.
4855::
4845::
4804:.
4749::
4719:.
4677:.
4603:)
4462:)
4457:0
4453:y
4449:,
4444:0
4440:x
4436:(
4433:=
4430:P
4409:n
4405:)
4403:n
4399:n
4397:(
4393:n
4389:n
4383:n
4344:)
4338:(
4333:)
4329:(
4315:.
4132:9
4118:8
4104:7
4090:6
4076:5
4062:4
4048:3
4034:2
3971:∞
3597:9
3584:8
3559:7
3541:6
3524:5
3507:4
3489:3
3471:2
3458:1
3254:.
3210:.
3205:i
3201:y
3195:1
3189:n
3184:0
3181:=
3178:i
3168:n
3165:1
3160:=
3155:y
3151:c
3129:,
3124:i
3120:x
3114:1
3108:n
3103:0
3100:=
3097:i
3087:n
3084:1
3079:=
3074:x
3070:c
3056:n
3046:n
3039:n
3036:(
3017:A
2994:.
2991:)
2986:i
2982:y
2976:1
2973:+
2970:i
2966:x
2957:1
2954:+
2951:i
2947:y
2941:i
2937:x
2933:(
2930:)
2925:1
2922:+
2919:i
2915:y
2911:+
2906:i
2902:y
2898:(
2893:1
2887:n
2882:0
2879:=
2876:i
2865:A
2862:6
2858:1
2853:=
2848:y
2844:C
2822:,
2819:)
2814:i
2810:y
2804:1
2801:+
2798:i
2794:x
2785:1
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2779:i
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2769:i
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2761:(
2758:)
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2750:+
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2743:x
2739:+
2734:i
2730:x
2726:(
2721:1
2715:n
2710:0
2707:=
2704:i
2693:A
2690:6
2686:1
2681:=
2676:x
2672:C
2614:.
2608:2
2602:n
2581:2
2575:n
2557:n
2554:=
2548:2
2542:n
2525:4
2519:2
2515:s
2511:n
2505:=
2500:n
2484:4
2478:2
2474:s
2470:n
2464:=
2461:A
2438:,
2425:s
2421:n
2402:n
2383:n
2366:n
2358:2
2354:R
2350:=
2345:n
2338:2
2321:2
2318:n
2308:2
2304:R
2300:=
2297:A
2280:R
2276:n
2269:a
2248:.
2245:r
2239:p
2230:2
2227:1
2221:=
2218:A
2205:p
2197:r
2178:n
2174:n
2146:A
2140:4
2132:2
2128:p
2110:p
2079:.
2076:)
2073:]
2070:)
2065:2
2059:n
2051:(
2040:1
2034:n
2030:a
2026:[
2021:2
2015:n
2011:a
2007:+
2001:+
1992:]
1989:)
1984:2
1978:n
1970:+
1964:+
1959:2
1951:(
1940:1
1934:n
1930:a
1926:+
1920:+
1917:)
1912:3
1904:+
1899:2
1891:(
1880:4
1876:a
1872:+
1869:)
1864:2
1856:(
1845:3
1841:a
1837:[
1832:2
1828:a
1824:+
1815:]
1812:)
1807:2
1801:n
1793:+
1787:+
1782:2
1774:+
1769:1
1761:(
1750:1
1744:n
1740:a
1736:+
1730:+
1727:)
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1714:+
1709:1
1701:(
1690:3
1686:a
1682:+
1679:)
1674:1
1666:(
1655:2
1651:a
1647:[
1642:1
1638:a
1634:(
1629:2
1626:1
1621:=
1618:A
1600:n
1598:θ
1594:2
1591:θ
1587:1
1584:θ
1575:n
1573:a
1569:2
1566:a
1562:1
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1555:A
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1530:x
1510:.
1507:)
1502:j
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1456:i
1452:y
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1435:(
1413:j
1410:,
1407:i
1403:Q
1379:,
1374:|
1366:1
1363:+
1360:j
1357:,
1354:1
1351:+
1348:i
1344:Q
1336:j
1333:,
1330:1
1327:+
1324:i
1320:Q
1310:1
1307:+
1304:j
1301:,
1298:i
1294:Q
1286:j
1283:,
1280:i
1276:Q
1269:|
1262:1
1256:n
1251:0
1248:=
1245:j
1235:1
1229:n
1224:0
1221:=
1218:i
1210:=
1205:2
1201:A
1168:,
1163:0
1159:y
1155:=
1150:n
1146:y
1135:0
1131:x
1127:=
1122:n
1118:x
1108:)
1103:i
1099:y
1093:1
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1074:1
1071:+
1068:i
1064:y
1058:i
1054:x
1050:(
1045:1
1039:n
1034:0
1031:=
1028:i
1018:2
1015:1
1010:=
1007:A
972:)
970:0
967:y
963:0
960:x
955:n
953:y
948:n
946:x
944:(
930:)
925:1
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906:1
900:n
896:x
892:(
889:,
883:,
880:)
875:1
871:y
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862:1
858:x
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851:,
848:)
843:0
839:y
835:,
830:0
826:x
822:(
787:d
775:d
771:n
763:n
734:p
730:)
727:q
724:2
718:p
715:(
685:p
681:)
678:q
675:2
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666:(
649:q
645:p
628:q
625:p
591:n
555:)
548:n
545:2
536:1
532:(
521:n
515:n
513:(
509:n
505:n
496:n
494:(
486:π
482:n
480:(
476:n
446:n
441:n
420:L
366:.
356:.
126:n
120:n
75:/
72:n
69:ɒ
66:ɡ
63:ɪ
60:l
57:ɒ
54:p
51:ˈ
48:/
44:(
23:.
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