930:
1840:
38:
1692:
549:
1796:
1787:
1832:
922:
1814:
1805:
1681:
1778:
213:
1731:
arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in
Euclidean solid geometry is a non-Euclidean surface in the sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry.
655:
through each pair of opposite sides. This definition includes both right-angled rectangles and crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the
1461:
1546:, with two ways of being folded along a line through its center such that the area of overlap is minimized and each area yields a different shape – a triangle and a pentagon. The unique ratio of side lengths is
671:
Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise
538:
1355:
1579:
251:
is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an
2419:
1131:
425:
1078:
1850:
A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is
112:
102:
1012:
1161:
2249:
with interactive animation illustrating a rectangle that becomes a 'crossed rectangle', making a good case for regarding a 'crossed rectangle' as a type of rectangle.
107:
1395:
955:
1544:
1524:
1504:
1484:
975:
1598:(self-intersecting) consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is a
1754:
is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length.
1743:
is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length.
1711:
interior defined as a linear combination of the four vertices, creating a saddle surface. This example shows 4 blue edges of the rectangle, and two
2087:
Zalman
Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, pp. 34–36
2242:
1220:
states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle.
449:
1606:
as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.
185:. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a
2182:
1959:
1253:
1217:
2533:
2494:
2426:
2440:
1549:
1911:
U+25AC ▬ BLACK RECTANGLE U+25AD ▭ WHITE RECTANGLE U+25AE ▮ BLACK VERTICAL RECTANGLE U+25AF ▯ WHITE VERTICAL RECTANGLE
1881:
sides if and only if it is tileable by a finite number of unequal squares. The same is true if the tiles are unequal isosceles
2120:
1888:
The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular
2664:
1590:
2092:
255:, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as
3194:
2344:
1086:
1870:. The lowest number of squares need for a perfect tiling of a rectangle is 9 and the lowest number needed for a
2018:
1878:
891:
371:
120:
2528:"Sequence A219766 (Number of nonsquare simple perfect squared rectangles of order n up to symmetry)"
1023:
1858:
and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is
267:, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.
1843:
Lowest-order perfect squared square (1) and the three smallest perfect squared squares (2–4) –
2787:
2767:
3199:
2762:
2719:
2694:
2023:
Philosophical
Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences
94:
2462:
1377:
2239:
2822:
1602:
which consists of two opposite sides of a rectangle along with the two diagonals. It has the same
2747:
1866:. A database of all known perfect rectangles, perfect squares and related shapes can be found at
1643:
of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.
1665:
A rectangle and a crossed rectangle are quadrilaterals with the following properties in common:
987:
2772:
2657:
1700:
1139:
895:
601:
3173:
3113:
2752:
2179:
2110:
1966:
1855:
1177:
739:
334:
1675:
It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
3057:
2827:
2757:
2699:
2260:
2066:
2030:
1224:
940:
929:
8:
3163:
3138:
3108:
3103:
3062:
2777:
2574:
1871:
1818:
1747:
1651:
884:
743:
709:
652:
638:
619:
597:
264:
2572:
R. Sprague (1940). "Ũber die
Zerlegung von Rechtecken in lauter verschiedene Quadrate".
2341:"Sequence A366185 (Decimal expansion of the real root of the quintic equation
2034:
1882:
3168:
2709:
2591:
2318:
2283:
2155:
2141:
Gerard Venema, "Exploring
Advanced Euclidean Geometry with GeoGebra", MAA, 2013, p. 56.
2070:
2054:
1720:
1704:
1603:
1529:
1509:
1489:
1469:
960:
870:
The figure formed by joining, in order, the midpoints of the sides of a rectangle is a
677:
673:
256:
170:
2548:
903:
84:
3148:
2742:
2650:
2613:
2595:
2322:
2216:
2116:
2088:
2074:
2046:
1736:
1622:
1164:
810:
724:
260:
252:
204:
125:
74:
20:
1989:
1456:{\displaystyle 0.5{\text{ × Area}}(R)\leq {\text{Area}}(C)\leq 2{\text{ × Area}}(r)}
2677:
2583:
2503:
2470:
2310:
2279:
2275:
2106:
2038:
1925:
1839:
1365:
661:
2474:
3143:
3123:
3118:
3088:
2807:
2782:
2714:
2457:
2246:
2186:
2062:
1708:
1640:
735:
578:
144:
70:
2636:
2005:
1662:
is 720°, allowing for internal angles to appear on the outside and exceed 180°.
3153:
3133:
3098:
3093:
2724:
2704:
2616:
1930:
1863:
1659:
728:
717:
694:
642:: The whole interior is visible from a single point, without crossing any edge.
630:
612:
294:
287:
148:
140:
1892:, allowing all rotations and reflections. There are also tilings by congruent
660:
bisector of those sides, but, in the case of the crossed rectangle, the first
3188:
3128:
2979:
2872:
2792:
2734:
2587:
2050:
1206:
1196:
1192:
910:
657:
615:
582:
574:
553:
302:
290:
186:
178:
59:
51:
2630:
37:
3158:
3028:
2984:
2948:
2938:
2933:
2549:"Squared Squares; Perfect Simples, Perfect Compounds and Imperfect Simples"
2508:
2489:
2455:
2042:
1935:
1763:
1728:
1691:
909:
Two rectangles, neither of which will fit inside the other, are said to be
816:
755:
698:
189:
containing a right angle. A rectangle with four sides of equal length is a
156:
2104:
274:
problems, such as tiling the plane by rectangles or tiling a rectangle by
3067:
2974:
2953:
2943:
2523:
2336:
1629:
651:
De
Villiers defines a rectangle more generally as any quadrilateral with
306:
182:
1862:. In a perfect (or imperfect) triangled rectangle the triangles must be
1795:
1786:
196:
3072:
2928:
2918:
2802:
2490:"On the Dissection of Rectangles into Right-Angled Isosceles Triangles"
2314:
2287:
548:
3047:
3037:
3014:
3004:
2994:
2923:
2832:
2621:
2487:
2058:
2021:; Longuet-Higgins, M.S.; Miller, J.C.P. (1954). "Uniform polyhedra".
1893:
1889:
1831:
1767:
1680:
1618:
1210:
1199:
1181:
1018:
921:
676:
and crossed isosceles trapezia (crossed quadrilaterals with the same
589:
557:
346:
63:
55:
533:{\displaystyle {\tfrac {1}{2}}{\sqrt {(a^{2}+c^{2})(b^{2}+d^{2})}}.}
3052:
3042:
2999:
2958:
2887:
2877:
2867:
2686:
2301:
Lassak, M. (1993). "Approximation of convex bodies by rectangles".
1813:
1804:
899:
665:
323:
312:
2642:
1350:{\displaystyle \displaystyle (AP)^{2}+(CP)^{2}=(BP)^{2}+(DP)^{2}.}
894:
consisting, for example, of three for position (comprising two of
3009:
2989:
2902:
2897:
2892:
2882:
2857:
2812:
2673:
1905:
1614:
871:
759:
275:
161:
2488:
J.D. Skinner II; C.A.B. Smith & W.T. Tutte (November 2000).
345:
a quadrilateral where the two diagonals are equal in length and
2817:
2456:
R.L. Brooks; C.A.B. Smith; A.H. Stone & W.T. Tutte (1940).
1920:
1826:
1777:
933:
The area of a rectangle is the product of the length and width.
702:
605:
561:
200:
191:
2202:
de
Villiers, Michael, "Generalizing Van Aubel Using Duality",
1715:
diagonals, all being diagonal of the cuboid rectangular faces.
2862:
1712:
713:
588:
A parallelogram is a special case of a trapezium (known as a
239:
224:
212:
2017:
890:
A rectangle in the plane can be defined by five independent
2527:
2340:
1960:"A Miscellany of Extracts from a Dictionary of Mathematics"
1867:
1626:
1574:{\displaystyle \displaystyle {\frac {a}{b}}=0.815023701...}
1185:
1180:
for rectangles states that among all rectangles of a given
982:
2258:
2156:"Five Proofs of an Area Characterization of Rectangles"
2522:
2335:
454:
376:
2347:
1553:
1552:
1532:
1512:
1492:
1472:
1398:
1257:
1256:
1142:
1089:
1026:
990:
963:
943:
452:
374:
2611:
2413:
1573:
1538:
1518:
1498:
1478:
1455:
1392:and the positive homothety ratio is at most 2 and
1349:
1155:
1125:
1072:
1006:
969:
949:
532:
419:
2261:"An Unexpected Maximum in a Family of Rectangles"
1654:if right and left turns are allowed. As with any
1632:that is twisted can take the shape of a bow tie.
3186:
2153:
2575:Journal für die reine und angewandte Mathematik
2658:
430:a convex quadrilateral with successive sides
352:a convex quadrilateral with successive sides
2180:An Extended Classification of Quadrilaterals
1827:Squared, perfect, and other tiled rectangles
925:The formula for the perimeter of a rectangle
749:
2458:"The dissection of rectangles into squares"
2414:{\displaystyle \ x^{5}+3x^{4}+4x^{3}+x-1=0}
2149:
2147:
1938:(includes a rectangle with rounded corners)
1466:There exists a unique rectangle with sides
1126:{\displaystyle d={\sqrt {\ell ^{2}+w^{2}}}}
2665:
2651:
2571:
2259:Hall, Leon M. & Robert P. Roe (1998).
1218:Japanese theorem for cyclic quadrilaterals
2534:On-Line Encyclopedia of Integer Sequences
2507:
2495:Journal of Combinatorial Theory, Series B
2427:On-Line Encyclopedia of Integer Sequences
1965:. Oxford University Press. Archived from
1874:is 21, found in 1978 by computer search.
1621:, sometimes called an "angular eight". A
1152:
1069:
1003:
420:{\displaystyle {\tfrac {1}{4}}(a+c)(b+d)}
2631:Definition and properties of a rectangle
2144:
1838:
1830:
1690:
1073:{\displaystyle P=2\ell +2w=2(\ell +w)\,}
928:
920:
646:
568:
547:
2451:
2449:
2240:Cyclic Quadrilateral Incentre-Rectangle
2189:(An excerpt from De Villiers, M. 1996.
2008:. Icoachmath.com. Retrieved 2011-11-13.
1762:The rectangle is used in many periodic
247:
151:Opposite angles and sides are congruent
3187:
2300:
2191:Some Adventures in Euclidean Geometry.
1957:
1672:The two diagonals are equal in length.
552:A rectangle is a special case of both
342:a quadrilateral with four right angles
2646:
2612:
1584:
843:Two axes of symmetry bisect opposite
836:Two axes of symmetry bisect opposite
634:: The boundary does not cross itself.
234:(as an adjective, right, proper) and
2446:
1953:
1951:
906:), and one for overall size (area).
281:
16:Quadrilateral with four right angles
2672:
2029:(916). The Royal Society: 401–450.
2006:Oblong – Geometry – Math Dictionary
1686:
1669:Opposite sides are equal in length.
823:Its centre is equidistant from its
808:Its centre is equidistant from its
573:A rectangle is a special case of a
13:
1247:on the same plane of a rectangle:
1227:states that with vertices denoted
1191:The midpoints of the sides of any
887:: its sides meet at right angles.
727:: all corners lie within the same
271:
223:The word rectangle comes from the
14:
3211:
2605:
1948:
1727:is a figure whose four edges are
668:for either side that it bisects.
564:is a special case of a rectangle.
543:
2193:University of Durban-Westville.)
1812:
1803:
1794:
1785:
1776:
1757:
1679:
878:
297:it is any one of the following:
270:Rectangles are involved in many
211:
110:
105:
100:
36:
2565:
2541:
2516:
2481:
2434:
2329:
2294:
2252:
2233:
2209:
2206:73 (4), Oct. 2000, pp. 303–307.
2196:
2019:Coxeter, Harold Scott MacDonald
1908:code points depict rectangles:
762:, as shown in the table below.
618:which has at least one pair of
577:in which each pair of adjacent
2443:. (PDF). Retrieved 2011-11-13.
2280:10.1080/0025570X.1998.11996653
2173:
2135:
2112:Methods for Euclidean Geometry
2098:
2081:
2011:
1999:
1983:
1845:all are simple squared squares
1835:A perfect rectangle of order 9
1770:, for example, these tilings:
1450:
1444:
1430:
1424:
1413:
1407:
1334:
1324:
1312:
1302:
1290:
1280:
1268:
1258:
1066:
1054:
522:
496:
493:
467:
414:
402:
399:
387:
1:
2475:10.1215/S0012-7094-40-00718-9
1942:
1184:, the square has the largest
859:Diagonals intersect at equal
683:
1774:
596:pairs of opposite sides are
339:an equiangular quadrilateral
230:, which is a combination of
203:rectangle. A rectangle with
199:" is used to refer to a non-
7:
2639:with interactive animation.
2633:with interactive animation.
2105:Owen Byer; Felix Lazebnik;
1958:Tapson, Frank (July 1999).
1914:
1171:
916:
746:of order 2 (through 180°).
688:
592:in North America) in which
10:
3216:
2524:Sloane, N. J. A.
2337:Sloane, N. J. A.
1899:
1699:has 4 nonplanar vertices,
1613:is sometimes likened to a
1007:{\displaystyle A=\ell w\,}
937:If a rectangle has length
625:A convex quadrilateral is
19:For the record label, see
18:
3081:
3027:
2967:
2911:
2850:
2841:
2733:
2685:
2154:Josefsson Martin (2013).
1156:{\displaystyle \ell =w\,}
1083:each diagonal has length
750:Rectangle-rhombus duality
155:
136:
119:
93:
83:
69:
47:
35:
30:
2588:10.1515/crll.1940.182.60
1872:perfect tilling a square
680:as isosceles trapezia).
171:Euclidean plane geometry
3195:Types of quadrilaterals
1996:. Retrieved 2011-11-13.
1388:is circumscribed about
852:Diagonals are equal in
95:Coxeter–Dynkin diagrams
2509:10.1006/jctb.2000.1987
2415:
2043:10.1098/rsta.1954.0003
1990:"Definition of Oblong"
1847:
1836:
1716:
1575:
1540:
1520:
1500:
1480:
1457:
1360:For every convex body
1351:
1157:
1127:
1074:
1008:
971:
951:
934:
926:
716:are equal (each of 90
565:
534:
421:
2416:
2115:. MAA. pp. 53–.
1842:
1834:
1694:
1656:crossed quadrilateral
1611:crossed quadrilateral
1600:crossed quadrilateral
1576:
1541:
1521:
1501:
1481:
1458:
1364:in the plane, we can
1352:
1178:isoperimetric theorem
1163:, the rectangle is a
1158:
1128:
1075:
1009:
972:
952:
950:{\displaystyle \ell }
932:
924:
740:reflectional symmetry
647:Alternative hierarchy
569:Traditional hierarchy
551:
535:
422:
311:a parallelogram with
2898:Nonagon/Enneagon (9)
2828:Tangential trapezoid
2441:Stars: A Second Look
2345:
2268:Mathematics Magazine
2204:Mathematics Magazine
1752:hyperbolic rectangle
1550:
1530:
1510:
1490:
1470:
1396:
1254:
1225:British flag theorem
1140:
1087:
1024:
988:
961:
941:
758:of a rectangle is a
450:
372:
210:would be denoted as
3010:Megagon (1,000,000)
2778:Isosceles trapezoid
2637:Area of a rectangle
2303:Geometriae Dedicata
2163:Forum Geometricorum
2107:Deirdre L. Smeltzer
2035:1954RSPTA.246..401C
1819:Herringbone pattern
1748:hyperbolic geometry
1725:spherical rectangle
1703:from vertices of a
885:rectilinear polygon
744:rotational symmetry
132:), , (*22), order 4
89:{ } × { }
2980:Icositetragon (24)
2614:Weisstein, Eric W.
2537:. OEIS Foundation.
2430:. OEIS Foundation.
2411:
2315:10.1007/BF01263495
2245:2011-09-28 at the
2185:2019-12-30 at the
2109:(19 August 2010).
1848:
1837:
1741:elliptic rectangle
1721:spherical geometry
1717:
1705:rectangular cuboid
1650:may be considered
1635:The interior of a
1604:vertex arrangement
1585:Crossed rectangles
1571:
1570:
1536:
1516:
1496:
1476:
1453:
1347:
1346:
1202:form a rectangle.
1153:
1123:
1070:
1004:
967:
947:
935:
927:
902:), one for shape (
892:degrees of freedom
827:, hence it has an
723:It is isogonal or
678:vertex arrangement
674:isosceles trapezia
664:is not an axis of
566:
530:
463:
417:
385:
305:with at least one
3200:Elementary shapes
3182:
3181:
3023:
3022:
3000:Myriagon (10,000)
2985:Triacontagon (30)
2949:Heptadecagon (17)
2939:Pentadecagon (15)
2934:Tetradecagon (14)
2873:Quadrilateral (4)
2743:Antiparallelogram
2350:
2122:978-0-88385-763-2
1854:if the tiles are
1824:
1823:
1737:elliptic geometry
1658:, the sum of its
1648:crossed rectangle
1637:crossed rectangle
1623:three-dimensional
1562:
1539:{\displaystyle b}
1519:{\displaystyle a}
1499:{\displaystyle b}
1479:{\displaystyle a}
1442:
1441: × Area
1422:
1405:
1404: × Area
1121:
970:{\displaystyle w}
883:A rectangle is a
867:
866:
814:, hence it has a
725:vertex-transitive
712:: all its corner
611:A trapezium is a
525:
462:
384:
282:Characterizations
253:antiparallelogram
248:crossed rectangle
167:
166:
21:Rectangle (label)
3207:
2995:Chiliagon (1000)
2975:Icositrigon (23)
2954:Octadecagon (18)
2944:Hexadecagon (16)
2848:
2847:
2667:
2660:
2653:
2644:
2643:
2627:
2626:
2600:
2599:
2569:
2563:
2562:
2560:
2559:
2553:www.squaring.net
2545:
2539:
2538:
2520:
2514:
2513:
2511:
2485:
2479:
2478:
2453:
2444:
2438:
2432:
2431:
2420:
2418:
2417:
2412:
2392:
2391:
2376:
2375:
2360:
2359:
2348:
2333:
2327:
2326:
2298:
2292:
2291:
2265:
2256:
2250:
2237:
2231:
2230:
2228:
2227:
2213:
2207:
2200:
2194:
2177:
2171:
2170:
2160:
2151:
2142:
2139:
2133:
2132:
2130:
2129:
2102:
2096:
2085:
2079:
2078:
2015:
2009:
2003:
1997:
1987:
1981:
1980:
1978:
1977:
1971:
1964:
1955:
1926:Golden rectangle
1877:A rectangle has
1846:
1816:
1807:
1798:
1789:
1780:
1773:
1772:
1707:, with a unique
1697:saddle rectangle
1687:Other rectangles
1683:
1580:
1578:
1577:
1572:
1563:
1555:
1545:
1543:
1542:
1537:
1525:
1523:
1522:
1517:
1505:
1503:
1502:
1497:
1485:
1483:
1482:
1477:
1462:
1460:
1459:
1454:
1443:
1440:
1423:
1420:
1406:
1403:
1356:
1354:
1353:
1348:
1342:
1341:
1320:
1319:
1298:
1297:
1276:
1275:
1243:, for any point
1213:is a rectangle.
1162:
1160:
1159:
1154:
1132:
1130:
1129:
1124:
1122:
1120:
1119:
1107:
1106:
1097:
1079:
1077:
1076:
1071:
1013:
1011:
1010:
1005:
976:
974:
973:
968:
956:
954:
953:
948:
765:
764:
701:lie on a single
653:axes of symmetry
622:opposite sides.
539:
537:
536:
531:
526:
521:
520:
508:
507:
492:
491:
479:
478:
466:
464:
455:
426:
424:
423:
418:
386:
377:
318:a parallelogram
215:
115:
114:
113:
109:
108:
104:
103:
40:
28:
27:
3215:
3214:
3210:
3209:
3208:
3206:
3205:
3204:
3185:
3184:
3183:
3178:
3077:
3031:
3019:
2963:
2929:Tridecagon (13)
2919:Hendecagon (11)
2907:
2843:
2837:
2808:Right trapezoid
2729:
2681:
2671:
2608:
2603:
2570:
2566:
2557:
2555:
2547:
2546:
2542:
2521:
2517:
2486:
2482:
2454:
2447:
2439:
2435:
2387:
2383:
2371:
2367:
2355:
2351:
2346:
2343:
2342:
2334:
2330:
2299:
2295:
2263:
2257:
2253:
2247:Wayback Machine
2238:
2234:
2225:
2223:
2215:
2214:
2210:
2201:
2197:
2187:Wayback Machine
2178:
2174:
2158:
2152:
2145:
2140:
2136:
2127:
2125:
2123:
2103:
2099:
2086:
2082:
2016:
2012:
2004:
2000:
1988:
1984:
1975:
1973:
1969:
1962:
1956:
1949:
1945:
1917:
1912:
1902:
1883:right triangles
1864:right triangles
1844:
1829:
1817:
1808:
1799:
1790:
1781:
1760:
1709:minimal surface
1689:
1660:interior angles
1641:polygon density
1587:
1554:
1551:
1548:
1547:
1531:
1528:
1527:
1511:
1508:
1507:
1491:
1488:
1487:
1471:
1468:
1467:
1439:
1419:
1402:
1397:
1394:
1393:
1337:
1333:
1315:
1311:
1293:
1289:
1271:
1267:
1255:
1252:
1251:
1174:
1141:
1138:
1137:
1115:
1111:
1102:
1098:
1096:
1088:
1085:
1084:
1025:
1022:
1021:
989:
986:
985:
962:
959:
958:
942:
939:
938:
919:
881:
874:and vice versa.
752:
693:A rectangle is
691:
686:
649:
571:
546:
516:
512:
503:
499:
487:
483:
474:
470:
465:
453:
451:
448:
447:
375:
373:
370:
369:
315:of equal length
293:is a rectangle
284:
131:
111:
106:
101:
99:
85:Schläfli symbol
43:
24:
17:
12:
11:
5:
3213:
3203:
3202:
3197:
3180:
3179:
3177:
3176:
3171:
3166:
3161:
3156:
3151:
3146:
3141:
3136:
3134:Pseudotriangle
3131:
3126:
3121:
3116:
3111:
3106:
3101:
3096:
3091:
3085:
3083:
3079:
3078:
3076:
3075:
3070:
3065:
3060:
3055:
3050:
3045:
3040:
3034:
3032:
3025:
3024:
3021:
3020:
3018:
3017:
3012:
3007:
3002:
2997:
2992:
2987:
2982:
2977:
2971:
2969:
2965:
2964:
2962:
2961:
2956:
2951:
2946:
2941:
2936:
2931:
2926:
2924:Dodecagon (12)
2921:
2915:
2913:
2909:
2908:
2906:
2905:
2900:
2895:
2890:
2885:
2880:
2875:
2870:
2865:
2860:
2854:
2852:
2845:
2839:
2838:
2836:
2835:
2830:
2825:
2820:
2815:
2810:
2805:
2800:
2795:
2790:
2785:
2780:
2775:
2770:
2765:
2760:
2755:
2750:
2745:
2739:
2737:
2735:Quadrilaterals
2731:
2730:
2728:
2727:
2722:
2717:
2712:
2707:
2702:
2697:
2691:
2689:
2683:
2682:
2670:
2669:
2662:
2655:
2647:
2641:
2640:
2634:
2628:
2607:
2606:External links
2604:
2602:
2601:
2582:(182): 60–64.
2564:
2540:
2515:
2502:(2): 277–319.
2480:
2469:(1): 312–340.
2445:
2433:
2410:
2407:
2404:
2401:
2398:
2395:
2390:
2386:
2382:
2379:
2374:
2370:
2366:
2363:
2358:
2354:
2328:
2293:
2274:(4): 285–291.
2251:
2232:
2208:
2195:
2172:
2143:
2134:
2121:
2097:
2080:
2010:
1998:
1982:
1946:
1944:
1941:
1940:
1939:
1933:
1931:Hyperrectangle
1928:
1923:
1916:
1913:
1910:
1904:The following
1901:
1898:
1828:
1825:
1822:
1821:
1810:
1801:
1792:
1783:
1759:
1756:
1688:
1685:
1677:
1676:
1673:
1670:
1586:
1583:
1569:
1568:0.815023701...
1566:
1561:
1558:
1535:
1515:
1495:
1475:
1452:
1449:
1446:
1438:
1435:
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1047:
1044:
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1038:
1035:
1032:
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1002:
999:
996:
993:
966:
946:
918:
915:
880:
877:
876:
875:
865:
864:
857:
849:
848:
841:
833:
832:
821:
805:
804:
797:
789:
788:
781:
773:
772:
769:
751:
748:
729:symmetry orbit
690:
687:
685:
682:
648:
645:
644:
643:
635:
570:
567:
545:
544:Classification
542:
541:
540:
529:
524:
519:
515:
511:
506:
502:
498:
495:
490:
486:
482:
477:
473:
469:
461:
458:
446:whose area is
428:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
383:
380:
368:whose area is
350:
343:
340:
337:
316:
309:
295:if and only if
283:
280:
165:
164:
159:
153:
152:
138:
134:
133:
129:
123:
121:Symmetry group
117:
116:
97:
91:
90:
87:
81:
80:
77:
67:
66:
49:
45:
44:
41:
33:
32:
15:
9:
6:
4:
3:
2:
3212:
3201:
3198:
3196:
3193:
3192:
3190:
3175:
3174:Weakly simple
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3114:Infinite skew
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3086:
3084:
3080:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3035:
3033:
3030:
3029:Star polygons
3026:
3016:
3015:Apeirogon (∞)
3013:
3011:
3008:
3006:
3003:
3001:
2998:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2972:
2970:
2966:
2960:
2959:Icosagon (20)
2957:
2955:
2952:
2950:
2947:
2945:
2942:
2940:
2937:
2935:
2932:
2930:
2927:
2925:
2922:
2920:
2917:
2916:
2914:
2910:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2855:
2853:
2849:
2846:
2840:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2793:Parallelogram
2791:
2789:
2788:Orthodiagonal
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2768:Ex-tangential
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2740:
2738:
2736:
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2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
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2698:
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2690:
2688:
2684:
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2663:
2661:
2656:
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2649:
2648:
2645:
2638:
2635:
2632:
2629:
2624:
2623:
2618:
2615:
2610:
2609:
2597:
2593:
2589:
2585:
2581:
2578:(in German).
2577:
2576:
2568:
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2550:
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2519:
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2505:
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2484:
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2472:
2468:
2465:
2464:
2463:Duke Math. J.
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2101:
2094:
2093:1-59311-695-0
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2044:
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2036:
2032:
2028:
2024:
2020:
2014:
2007:
2002:
1995:
1991:
1986:
1972:on 2014-05-14
1968:
1961:
1954:
1952:
1947:
1937:
1934:
1932:
1929:
1927:
1924:
1922:
1919:
1918:
1909:
1907:
1897:
1895:
1891:
1886:
1884:
1880:
1879:commensurable
1875:
1873:
1869:
1865:
1861:
1857:
1853:
1841:
1833:
1820:
1815:
1811:
1809:Basket weave
1806:
1802:
1800:Basket weave
1797:
1793:
1791:Running bond
1788:
1784:
1782:Stacked bond
1779:
1775:
1771:
1769:
1766:patterns, in
1765:
1758:Tessellations
1755:
1753:
1749:
1744:
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1616:
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1601:
1597:
1596:quadrilateral
1594:
1593:
1582:
1567:
1564:
1559:
1556:
1533:
1526:is less than
1513:
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1464:
1447:
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1416:
1410:
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1387:
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1234:
1230:
1226:
1221:
1219:
1214:
1212:
1208:
1207:parallelogram
1203:
1201:
1198:
1197:perpendicular
1194:
1193:quadrilateral
1189:
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897:
893:
888:
886:
879:Miscellaneous
873:
869:
868:
862:
858:
855:
851:
850:
846:
842:
839:
835:
834:
830:
826:
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766:
763:
761:
757:
747:
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741:
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719:
715:
711:
706:
704:
700:
696:
681:
679:
675:
669:
667:
663:
659:
658:perpendicular
654:
641:
640:
636:
633:
632:
628:
627:
626:
623:
621:
617:
616:quadrilateral
614:
609:
607:
603:
599:
595:
591:
586:
584:
583:perpendicular
580:
576:
575:parallelogram
563:
559:
555:
554:parallelogram
550:
527:
517:
513:
509:
504:
500:
488:
484:
480:
475:
471:
459:
456:
445:
441:
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408:
405:
396:
393:
390:
381:
378:
367:
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348:
344:
341:
338:
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332:
328:
325:
321:
317:
314:
310:
308:
304:
303:parallelogram
300:
299:
298:
296:
292:
291:quadrilateral
289:
279:
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268:
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184:
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179:quadrilateral
176:
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118:
98:
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92:
88:
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82:
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72:
68:
65:
61:
60:parallelogram
57:
53:
52:quadrilateral
50:
46:
39:
34:
29:
26:
22:
2968:>20 sides
2903:Decagon (10)
2888:Heptagon (7)
2878:Pentagon (5)
2868:Triangle (3)
2797:
2763:Equidiagonal
2620:
2579:
2573:
2567:
2556:. Retrieved
2552:
2543:
2531:
2518:
2499:
2493:
2483:
2466:
2461:
2436:
2424:
2331:
2306:
2302:
2296:
2271:
2267:
2254:
2235:
2224:. Retrieved
2220:
2211:
2203:
2198:
2190:
2175:
2166:
2162:
2137:
2126:. Retrieved
2111:
2100:
2083:
2026:
2022:
2013:
2001:
1993:
1985:
1974:. Retrieved
1967:the original
1936:Superellipse
1903:
1887:
1876:
1868:squaring.net
1859:
1851:
1849:
1764:tessellation
1761:
1751:
1745:
1740:
1734:
1729:great circle
1724:
1718:
1696:
1678:
1664:
1655:
1647:
1645:
1636:
1634:
1625:rectangular
1610:
1608:
1599:
1595:
1591:
1588:
1465:
1389:
1385:
1381:
1376:such that a
1373:
1369:
1368:a rectangle
1361:
1359:
1244:
1240:
1236:
1232:
1228:
1222:
1215:
1204:
1190:
1175:
936:
911:incomparable
908:
904:aspect ratio
889:
882:
860:
853:
844:
837:
828:
824:
817:circumcircle
815:
809:
800:
793:
784:
777:
756:dual polygon
753:
733:
722:
707:
692:
670:
650:
637:
629:
624:
610:
593:
587:
572:
443:
439:
435:
431:
365:
361:
357:
353:
330:
326:
319:
285:
269:
246:
244:
235:
231:
227:
222:
217:
207:
195:. The term "
190:
183:right angles
174:
168:
157:Dual polygon
25:
3164:Star-shaped
3139:Rectilinear
3109:Equilateral
3104:Equiangular
3068:Hendecagram
2912:11–20 sides
2893:Octagon (8)
2883:Hexagon (6)
2858:Monogon (1)
2700:Equilateral
2617:"Rectangle"
2309:: 111–117.
2221:Math Is Fun
2217:"Rectangle"
1994:Math Is Fun
1894:polyaboloes
1890:polyominoes
1652:equiangular
1639:can have a
1209:with equal
898:and one of
896:translation
803:are equal.
796:are equal.
787:are equal.
780:are equal.
734:It has two
710:equiangular
639:Star-shaped
307:right angle
228:rectangulus
3189:Categories
3169:Tangential
3073:Dodecagram
2851:1–10 sides
2842:By number
2823:Tangential
2803:Right kite
2558:2021-09-26
2226:2024-03-22
2128:2011-11-13
1976:2013-06-20
1943:References
1701:alternated
1378:homothetic
957:and width
799:Alternate
792:Alternate
684:Properties
349:each other
265:hyperbolic
181:with four
137:Properties
3149:Reinhardt
3058:Enneagram
3048:Heptagram
3038:Pentagram
3005:65537-gon
2863:Digon (2)
2833:Trapezoid
2798:Rectangle
2748:Bicentric
2710:Isosceles
2687:Triangles
2622:MathWorld
2596:118088887
2400:−
2323:119508642
2075:202575183
2051:0080-4614
1860:imperfect
1768:brickwork
1619:butterfly
1434:≤
1417:≤
1211:diagonals
1200:diagonals
1182:perimeter
1144:ℓ
1100:ℓ
1058:ℓ
1037:ℓ
1019:perimeter
998:ℓ
945:ℓ
768:Rectangle
590:trapezoid
558:trapezoid
335:congruent
324:triangles
313:diagonals
257:spherical
175:rectangle
64:orthotope
56:trapezium
42:Rectangle
31:Rectangle
3124:Isotoxal
3119:Isogonal
3063:Decagram
3053:Octagram
3043:Hexagram
2844:of sides
2773:Harmonic
2674:Polygons
2243:Archived
2183:Archived
2169:: 17–21.
1915:See also
1506:, where
1366:inscribe
1172:Theorems
977:, then:
917:Formulae
900:rotation
829:incircle
811:vertices
771:Rhombus
689:Symmetry
666:symmetry
620:parallel
598:parallel
276:polygons
261:elliptic
205:vertices
145:isogonal
126:Dihedral
75:vertices
3144:Regular
3089:Concave
3082:Classes
2990:257-gon
2813:Rhombus
2753:Crossed
2526:(ed.).
2339:(ed.).
2288:2690700
2067:0062446
2031:Bibcode
1906:Unicode
1900:Unicode
1856:similar
1852:perfect
1615:bow tie
1592:crossed
1017:it has
981:it has
872:rhombus
760:rhombus
718:degrees
699:corners
236:angulus
162:rhombus
3154:Simple
3099:Cyclic
3094:Convex
2818:Square
2758:Cyclic
2720:Obtuse
2715:Kepler
2594:
2349:
2321:
2286:
2119:
2091:
2073:
2065:
2057:
2049:
1921:Cuboid
1239:, and
1165:square
861:angles
854:length
845:angles
801:angles
778:angles
714:angles
708:It is
703:circle
697:: all
695:cyclic
631:Simple
613:convex
606:length
562:square
347:bisect
322:where
288:convex
272:tiling
263:, and
232:rectus
216:
201:square
197:oblong
192:square
149:cyclic
141:convex
3129:Magic
2725:Right
2705:Ideal
2695:Acute
2592:S2CID
2319:S2CID
2284:JSTOR
2264:(PDF)
2159:(PDF)
2071:S2CID
2059:91532
2055:JSTOR
1970:(PDF)
1963:(PDF)
1739:, an
1713:green
1630:frame
1380:copy
1195:with
1136:when
1133:; and
838:sides
825:sides
794:sides
785:sides
736:lines
602:equal
579:sides
240:angle
225:Latin
177:is a
71:Edges
3159:Skew
2783:Kite
2678:List
2580:1940
2532:The
2425:The
2117:ISBN
2089:ISBN
2047:ISSN
1750:, a
1723:, a
1627:wire
1486:and
1421:Area
1223:The
1216:The
1186:area
1176:The
983:area
783:All
776:All
754:The
742:and
662:axis
600:and
594:both
560:. A
556:and
333:are
329:and
320:ABCD
218:ABCD
208:ABCD
173:, a
73:and
48:Type
2584:doi
2504:doi
2471:doi
2311:doi
2276:doi
2039:doi
2027:246
1746:In
1735:In
1719:In
1617:or
1400:0.5
1384:of
1372:in
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720:).
604:in
581:is
331:DCA
327:ABD
242:).
169:In
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