4025:
quadrilaterals having two sides parallel, subdividing these into the τραπέζιον ἰσοσκελὲς, isosceles trapezium, having the two non-parallel sides (and the angles at their bases) equal, and σκαληνὸν τραπέζιον, scalene trapezium, in which these sides and angles are unequal. For quadrilaterals having no sides parallel, Proclus introduced the name τραπέζοειδὲς TRAPEZOID. This nomenclature is retained in all the continental languages, and was universal in
England till late in the 18th century, when the application of the terms was transposed, so that the figure which Proclus and modern geometers of other nations call specifically a trapezium (F. trapèze, Ger. trapez, Du. trapezium, It. trapezio) became with most English writers a trapezoid, and the trapezoid of Proclus and other nations a trapezium. This changed sense of trapezoid is given in Hutton's Mathematical Dictionary, 1795, as 'sometimes' used -- he does not say by whom; but he himself unfortunately adopted and used it, and his Dictionary was doubtless the chief agent in its diffusion. Some geometers however continued to use the terms in their original senses, and since c 1875 this is the prevalent use.
3872:
310:
1207:
1045:
3819:
704:
560:
519:
491:
603:
473:
2970:
44:
2952:
4024:
With Euclid (c 300 B.C.) τραπέζιον included all quadrilateral figures except the square, rectangle, rhombus, and rhomboid; into the varieties of trapezia he did not enter. But
Proclus, who wrote Commentaries on the First Book of Euclid's Elements A.D. 450, retained the name τραπέζιον only for
2234:
3914:. Multiplexors are logic elements that select between multiple elements and produce a single output based on a select signal. Typical designs will employ trapezoids without specifically stating they are multiplexors as they are universally equivalent.
3842:
In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the top, in
Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually
3540:
2684:
2436:
4054:πέζα is said to be the Doric and Arcadic form of πούς 'foot', but recorded only in the sense 'instep ', whence the meaning 'edge, border'. τράπεζα 'table' is Homeric. Henry George Liddell, Robert Scott, Henry Stuart Jones,
2588:
3083:
3187:
1875:
321:
defined five types of quadrilateral, of which four had two sets of parallel sides (known in
English as square, rectangle, rhombus and rhomboid) and the last did not have two sets of parallel sides – a τραπέζια
3639:
4346:"Math Education Geometry Problem 747: Trapezoid, Diagonals, Parallel, Bases, Midpoint, Similarity, Harmonic Mean. Level: High School, Honors Geometry, College, Mathematics Education. Distance learning"
1343:
919:
3863:
is the problem of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection.
3808:
672:
one pair of parallel sides (the inclusive definition), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as
2001:
1570:
2089:
2653:
1428:
400:
in 1795 supported without explanation a transposition of the terms. This was reversed in
British English in about 1875, but it has been retained in American English to the present.
3697:
1640:
1717:
120:
1034:
695:) are trapezoids. Rectangles have mirror symmetry on mid-edges; rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices.
3367:
3342:
3317:
3292:
3267:
3242:
1930:
966:
1103:
1235:
1159:
1131:
1077:
2947:{\displaystyle K={\sqrt {{\frac {(ab^{2}-a^{2}b-ad^{2}+bc^{2})(ab^{2}-a^{2}b-ac^{2}+bd^{2})}{4(b-a)^{2}}}-\left({\frac {c^{2}+d^{2}-a^{2}-b^{2}}{4}}\right)^{2}}}.}
3454:
1201:
1180:
2261:
4141:
3545:
The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.
396:
All
European languages follow Proclus's structure as did English until the late 18th century, until an influential mathematical dictionary published by
1281:
The product of the areas of the two triangles formed by one diagonal equals the product of the areas of the two triangles formed by the other diagonal.
4345:
2458:
2982:
3089:
1755:
676:. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the
3571:
4037:
4063:
1445:
Twice the length of the bimedian connecting the midpoints of two opposite sides equals the sum of the lengths of the other sides.
1241:
Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:
4266:
3910:
In computer engineering, specifically digital logic and computer architecture, trapezoids are typically utilized to symbolize
758:
is a trapezoid where the base angles have the same measure. As a consequence the two legs are also of equal length and it has
4498:
4090:
3373:. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.
4576:
4416:
1301:
849:
4019:
A New
English Dictionary on Historical Principles: Founded Mainly on the Materials Collected by the Philological Society
4622:
403:
The following table compares usages, with the most specific definitions at the top to the most general at the bottom.
771:
is (under the inclusive definition) a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold
3732:
2229:{\displaystyle K={\frac {1}{2}}(a+b){\sqrt {c^{2}-{\frac {1}{4}}\left((b-a)+{\frac {c^{2}-d^{2}}{b-a}}\right)^{2}}}}
5157:
657:
one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Some sources use the term
1950:
1494:
3871:
2596:
2060:, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.
1374:
4392:"Centroid, Area, Moments of Inertia, Polar Moments of Inertia, & Radius of Gyration of a General Trapezoid"
259:) if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases. A
309:
4127:
3650:
1591:
4745:
4725:
4537:
4059:
1678:
969:
17:
85:
5152:
4720:
4677:
4652:
3827:
3557:) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance
2668:
975:
1439:
The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.
4780:
4582:
2660:
1723:
789:
3644:
The center of area divides this segment in the ratio (when taken from the short to the long side)
369:
4705:
4377:
3993:
3860:
1292:
of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation
4331:
3352:
3327:
3302:
3277:
3252:
3227:
4730:
4615:
4488:
1891:
1360:
The midpoints of two opposite sides of the trapezoid and the intersection of the diagonals are
927:
796:
264:
184:
4080:
3894:
and other descriptive disciplines in which a term for such shapes is necessary, terms such as
5131:
5071:
4710:
808:
4593:
1449:
Additionally, the following properties are equivalent, and each implies that opposite sides
1082:
5015:
4785:
4715:
4657:
4017:
3887:
3535:{\displaystyle {\frac {1}{FG}}={\frac {1}{2}}\left({\frac {1}{AB}}+{\frac {1}{DC}}\right).}
2664:
1250:
1213:
1137:
1109:
1055:
784:
346:
4183:
2452:
Another equivalent formula for the area, which more closely resembles Heron's formula, is
8:
5121:
5096:
5066:
5061:
5020:
4735:
4532:
3844:
800:
772:
759:
754:
241:
294:
is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids.
5126:
4667:
4439:
4317:
3891:
2446:
2431:{\displaystyle K={\frac {a+b}{4|b-a|}}{\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}.}
2039:
1186:
1165:
279:
4281:
4159:
677:
5106:
4700:
4608:
4545:
4494:
4242:
4086:
3848:
1206:
688:
661:
to describe trapezoids under the exclusive definition, analogous to uses of the word
68:
4548:
4635:
4431:
4372:
3935:
3823:
2043:
1737:
distance between the bases. In the case that the two bases have different lengths (
776:
724:
499:
opposite sides and angles equal to one another but not equilateral nor right-angled
194:
148:
4570:
4391:
779:
symmetry). It is possible for obtuse trapezoids or right trapezoids (rectangles).
5101:
5081:
5076:
5046:
4765:
4740:
4672:
3880:
2067:
derived the following formula for the area of a trapezoid with consecutive sides
2023:
1656:(also called the median or midline) of a trapezoid is the segment that joins the
804:
283:
233:
64:
4566:
5111:
5091:
5056:
5051:
4682:
4662:
4518:
4245:
4204:
1254:
1044:
397:
357:(τραπέζιον), divided into isosceles (equal legs) and scalene (unequal) trapezia
275:
129:
1442:
One bimedian divides the quadrilateral into two quadrilaterals of equal areas.
5146:
5086:
4937:
4830:
4750:
4692:
3956:
3929:
3831:
3554:
3437:
2672:
2656:
2031:
1734:
767:
762:. This is possible for acute trapezoids or right trapezoids (as rectangles).
647:
237:
57:
4562:
2064:
707:
Trapezoid special cases. The orange figures also qualify as parallelograms.
650:, which have two pairs of parallel sides, should be regarded as trapezoids.
5116:
4986:
4942:
4906:
4896:
4891:
2053:
1271:(this ratio is the same as that between the lengths of the parallel sides).
835:
can constitute the consecutive sides of a non-parallelogram trapezoid with
720:
4414:
3818:
703:
5025:
4932:
4911:
4901:
3911:
2048:
4463:
4079:
Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (5 April 2016).
2583:{\displaystyle K={\frac {a+b}{|b-a|}}{\sqrt {(s-b)(s-a)(s-b-c)(s-b-d)}}}
5030:
4886:
4876:
4760:
4443:
2035:
1264:
is equal to the angle between the opposite side and the same diagonal.
803:
plane, with two adjacent right angles, while it is a rectangle in the
5005:
4995:
4972:
4962:
4952:
4881:
4755:
4553:
4250:
3876:
3078:{\displaystyle p={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},}
2027:
1361:
692:
4435:
3182:{\displaystyle q={\sqrt {\frac {ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}}}
2018:
is the height (the perpendicular distance between these sides), and
1870:{\displaystyle h={\frac {\sqrt {(p-a)(p-b)(p-b-d)(p-b-c)}}{2|b-a|}}}
1749:
can be determined by the length of its four sides using the formula
559:
518:
5010:
5000:
4957:
4916:
4845:
4835:
4825:
4644:
4493:. Springer Science & Business Media. pp. 386, 1062, 1247.
4022:. Vol. X. Clarendon Press at Oxford. p. 286 (Trapezium).
3902:
commonly are useful in descriptions of particular organs or forms.
2958:
2057:
1657:
1433:
1275:
1261:
673:
490:
384:
138:
4600:
3938:, a polyhedron defined by two triangles and three trapezoid faces.
602:
472:
4967:
4947:
4860:
4855:
4850:
4840:
4815:
4770:
4631:
3923:
3634:{\displaystyle x={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).}
3211:
If the trapezoid is divided into four triangles by its diagonals
684:
389:
342:
4775:
4579:
at elsy.at: Animated course (construction, circumference, area)
3847:. This was the standard style for the doors and windows of the
2667:(inscribed in a circle). The formula is also a special case of
374:
318:
683:
Under the inclusive definition, all parallelograms (including
212:
172:
4820:
4332:
Mathematical achievements of pre-modern Indian mathematicians
2255:. This formula can be factored into a more symmetric version
1726:(the other bimedian divides the trapezoid into equal areas).
1582:
between the midpoints of the diagonals satisfies the equation
1268:
1246:
745:
on the other hand has one acute and one obtuse angle on each
27:
Convex quadrilateral with at least one pair of parallel sides
4415:
Tom M. Apostol and
Mamikon A. Mnatsakanian (December 2004).
4078:
2969:
43:
2663:, but it differs from it, in that a trapezoid might not be
379:
297:
221:
218:
203:
166:
78:
2441:
When one of the parallel sides has shrunk to a point (say
4520:
Common properties of trapezoids and convex quadrilaterals
160:
4543:
3957:"Trapezoid - math word definition - Math Open Reference"
641:
345:(AD 412 to 485) in his commentary on the first book of
2607:
90:
4464:"Machu Picchu Lost City of the Incas - Inca Geometry"
3735:
3653:
3574:
3457:
3355:
3330:
3305:
3280:
3255:
3230:
3092:
2985:
2687:
2599:
2461:
2264:
2092:
1953:
1894:
1758:
1681:
1660:
of the legs. It is parallel to the bases. Its length
1594:
1497:
1377:
1304:
1216:
1189:
1168:
1140:
1112:
1085:
1058:
978:
930:
853:
852:
251:
of the trapezoid. The other two sides are called the
215:
169:
88:
2026:
of the lengths of the two parallel sides. In 499 AD
1664:
is equal to the average of the lengths of the bases
1338:{\displaystyle {\sqrt {K}}={\sqrt {S}}+{\sqrt {T}},}
914:{\displaystyle \displaystyle |d-c|<|b-a|<d+c.}
209:
200:
197:
175:
163:
157:
154:
4381:, Mathematical Association of America, 2010, p. 55.
4074:
4072:
224:
206:
151:
4240:
4015:
3802:
3691:
3633:
3534:
3361:
3336:
3311:
3286:
3261:
3236:
3181:
3077:
2946:
2647:
2582:
2430:
2228:
1995:
1924:
1869:
1711:
1634:
1564:
1422:
1337:
1267:The diagonals cut each other in mutually the same
1229:
1195:
1174:
1153:
1125:
1097:
1071:
1028:
960:
913:
668:Others define a trapezoid as a quadrilateral with
653:Some define a trapezoid as a quadrilateral having
114:
3553:The center of area (center of mass for a uniform
5144:
4486:
4069:
2961:connecting the parallel sides bisects the area.
1722:The midsegment of a trapezoid is one of the two
1436:to 0, as do the cosines of the other two angles.
4282:"Area of Trapezoid Given Only the Side Lengths"
4038:"Euclid, Elements, book 1, type Def, number 22"
3803:{\displaystyle PQ={\frac {|AD+BC-AB-CD|}{2}}.}
3404:be the intersection of the diagonals, and let
2678:From Bretschneider's formula, it follows that
2659:of the trapezoid. (This formula is similar to
1274:The diagonals cut the quadrilateral into four
4616:
4312:
4310:
3981:Plane Euclidean Geometry: Theory and Problems
4517:D. Fraivert, A. Sigler and M. Stupel :
4480:
4308:
4306:
4304:
4302:
4300:
4298:
4296:
4294:
4292:
4290:
4236:
4234:
4232:
4230:
4228:
4226:
4224:
4222:
4220:
4184:"American School definition from "math.com""
1996:{\displaystyle K={\frac {a+b}{2}}\cdot h=mh}
1565:{\displaystyle p^{2}+q^{2}=c^{2}+d^{2}+2ab.}
1278:of which one opposite pair have equal areas.
1210:trapezoid/trapezium with opposing triangles
811:in the hyperbolic plane has 3 right angles.
734:has two adjacent acute angles on its longer
575:Two parallel sides, and no line of symmetry
4623:
4609:
2648:{\displaystyle s={\tfrac {1}{2}}(a+b+c+d)}
1423:{\displaystyle \sin A\sin C=\sin B\sin D.}
924:The quadrilateral is a parallelogram when
601:
558:
537:Two parallel sides, and a line of symmetry
517:
489:
471:
274:A trapezoid is usually considered to be a
4287:
4217:
3219:(as shown on the right), intersecting at
2056:the well-known formula for the area of a
1223:
1147:
1119:
1065:
814:
4587:Numerical Methods for Stem Undergraduate
3870:
3817:
1647:
1205:
1043:
702:
441:(Definitions 30-34, quoting Posidonius)
308:
4367:
4365:
4320:, Forum Geometricorum, 13 (2013) 23-35.
4160:"1913 American definition of trapezium"
4058:, Oxford, Clarendon Press (1940), s.v.
3905:
2014:are the lengths of the parallel sides,
372:means 'resembles'), in the same way as
326:literally 'table', itself from τετράς (
14:
5145:
4604:
4544:
4241:
4011:
4009:
4007:
3692:{\displaystyle {\frac {a+2b}{2a+b}}.}
2063:The 7th-century Indian mathematician
1635:{\displaystyle v={\frac {|a-b|}{2}}.}
642:Inclusive versus exclusive definition
4362:
4128:"Larousse definition for trapézoïde"
3979:A. D. Gardiner & C. J. Bradley,
3932:, also known as a trapezoidal number
3714:, and the angle bisectors to angles
3392:in sequence and have parallel sides
1039:
727:for estimating areas under a curve.
665:in some other mathematical objects.
4630:
4594:Numerical Methods with Applications
4487:John L. Capinera (11 August 2008).
3548:
1712:{\displaystyle m={\frac {a+b}{2}}.}
1432:The cosines of two adjacent angles
723:. Right trapezoids are used in the
646:There is some disagreement whether
24:
4511:
4408:
4202:
4004:
3926:, a solid having trapezoidal faces
3356:
3331:
3306:
3281:
3273:, and the product of the areas of
3256:
3231:
2968:
268:
247:The parallel sides are called the
115:{\displaystyle {\tfrac {a+b}{2}}h}
25:
5169:
4526:
4318:"Characterizations of trapezoids"
4164:Merriam-Webster Online Dictionary
3702:If the angle bisectors to angles
2976:The lengths of the diagonals are
1355:is the area of the quadrilateral.
1029:{\displaystyle |d-c|=|b-a|\neq 0}
799:is similar to a trapezoid in the
4417:"Figures Circumscribing Circles"
3376:Let the trapezoid have vertices
2052:(section 2.8). This yields as a
1888:are the lengths of the legs and
1367:The angles in the quadrilateral
972:(which is not a trapezoid) when
698:
481:equilateral but not right-angled
317:The ancient Greek mathematician
263:is a trapezoid with no sides of
193:
147:
42:
4456:
4384:
4338:
4323:
4271:
4259:
4196:
4176:
4152:
3837:
3813:
1260:The angle between a side and a
353:one pair of parallel sides – a
4378:Methods for Euclidean Geometry
4371:Owen Byer, Felix Lazebnik and
4134:
4099:
4048:
4030:
3986:
3973:
3949:
3787:
3749:
2851:
2838:
2830:
2766:
2763:
2699:
2642:
2618:
2575:
2557:
2554:
2536:
2533:
2521:
2518:
2506:
2497:
2483:
2445:= 0), this formula reduces to
2420:
2396:
2393:
2369:
2366:
2342:
2339:
2312:
2303:
2289:
2167:
2155:
2121:
2109:
1860:
1846:
1837:
1819:
1816:
1798:
1795:
1783:
1780:
1768:
1619:
1605:
1253:, that is, they add up to 180
1016:
1002:
994:
980:
891:
877:
869:
855:
368:, literally 'trapezium-like' (
334:) 'foot; end, border, edge').
267:measure, in contrast with the
13:
1:
4573:(with interactive animations)
4424:American Mathematical Monthly
4125:, Ukrainian "трапеція", e.g.
3942:
1745:), the height of a trapezoid
4591:Autar Kaw and E. Eric Kalu,
2964:
2449:for the area of a triangle.
1049:general trapezoid/trapezium:
313:Hutton's definitions in 1795
290:is a convex trapezoid, then
7:
4538:Encyclopedia of Mathematics
4016:James A. H. Murray (1926).
3917:
3854:
1944:of a trapezoid is given by
970:ex-tangential quadrilateral
788:is a trapezoid that has an
445:Euclid / Proclus definition
10:
5174:
4490:Encyclopedia of Entomology
4205:"History and Nomenclature"
4085:. CRC Press. p. 286.
3866:
3828:Metropolitan Museum of Art
3362:{\displaystyle \triangle }
3337:{\displaystyle \triangle }
3312:{\displaystyle \triangle }
3287:{\displaystyle \triangle }
3262:{\displaystyle \triangle }
3237:{\displaystyle \triangle }
2046:, used this method in the
2038:from the classical age of
678:taxonomy of quadrilaterals
5039:
4985:
4925:
4869:
4808:
4799:
4691:
4643:
4577:Trapezoid (North America)
4335:, Elsevier, 2012, p. 156.
3994:"Types of Quadrilaterals"
3875:Example of a trapeziform
1925:{\displaystyle p=a+b+c+d}
961:{\displaystyle d-c=b-a=0}
523:
511:
506:
501:
495:
483:
477:
465:
460:
420:
417:
125:
77:
63:
53:
41:
34:
4146:www.chambersharrap.co.uk
4082:The Symmetries of Things
3208:are the trapezoid legs.
621:Irregular quadrilateral/
5158:Types of quadrilaterals
4268:. Retrieved 2012-02-24.
4056:A Greek-English Lexicon
3861:crossed ladders problem
2669:Bretschneider's formula
1935:
1237:formed by the diagonals
502:Rhomboid/Parallelogram
496:ῥομβοειδὲς (rhomboides)
4142:"chambersharrap.co.uk"
3883:
3834:
3804:
3693:
3635:
3536:
3363:
3338:
3313:
3288:
3263:
3238:
3200:is the long base, and
3183:
3079:
2973:
2948:
2649:
2584:
2432:
2230:
1997:
1926:
1871:
1713:
1636:
1566:
1461:The consecutive sides
1424:
1339:
1238:
1231:
1203:
1197:
1176:
1155:
1127:
1099:
1098:{\displaystyle a<b}
1073:
1030:
962:
915:
815:Condition of existence
797:Saccheri quadrilateral
717:right-angled trapezoid
708:
484:Rhombus/Parallelogram
412:Sets of parallel sides
314:
185:North American English
116:
48:Trapezoid or trapezium
4571:Median of a trapezoid
4042:www.perseus.tufts.edu
3998:Basic-mathematics.com
3874:
3821:
3805:
3694:
3636:
3561:from the longer side
3537:
3364:
3339:
3314:
3289:
3264:
3239:
3184:
3080:
2972:
2949:
2661:Brahmagupta's formula
2650:
2585:
2433:
2231:
1998:
1927:
1872:
1733:(or altitude) is the
1714:
1648:Midsegment and height
1637:
1567:
1425:
1340:
1232:
1230:{\displaystyle S,\,T}
1209:
1198:
1177:
1156:
1154:{\displaystyle q,\,p}
1128:
1126:{\displaystyle c,\,d}
1100:
1074:
1072:{\displaystyle a,\,b}
1047:
1031:
963:
916:
809:Lambert quadrilateral
706:
527:τραπέζιον ἰσοσκελὲς (
312:
282:, but there are also
240:that has one pair of
117:
4856:Nonagon/Enneagon (9)
4786:Tangential trapezoid
4563:Trapezoid definition
4105:For example: French
3983:, UKMT, 2005, p. 34.
3906:Computer engineering
3845:isosceles trapezoids
3733:
3651:
3572:
3455:
3353:
3328:
3323:is equal to that of
3303:
3278:
3253:
3248:is equal to that of
3228:
3090:
2983:
2685:
2597:
2459:
2262:
2090:
1951:
1892:
1756:
1679:
1592:
1495:
1485:satisfy the equation
1375:
1302:
1245:It has two adjacent
1214:
1187:
1166:
1138:
1110:
1083:
1056:
976:
928:
850:
785:tangential trapezoid
565:τραπέζιον σκαληνὸν (
418:Original terminology
360:no parallel sides –
86:
4968:Megagon (1,000,000)
4736:Isosceles trapezoid
4567:Area of a trapezoid
3961:www.mathopenref.com
3223:, then the area of
3196:is the short base,
843:parallel only when
773:rotational symmetry
760:reflection symmetry
755:isosceles trapezoid
719:) has two adjacent
524:τραπέζια (trapezia)
421:Modern terminology
341:were introduced by
4938:Icositetragon (24)
4546:Weisstein, Eric W.
4329:T. K. Puttaswamy,
4316:Martin Josefsson,
4243:Weisstein, Eric W.
4203:Michon, Gérard P.
3884:
3835:
3800:
3689:
3631:
3532:
3359:
3334:
3309:
3284:
3259:
3234:
3179:
3075:
2974:
2944:
2645:
2616:
2580:
2428:
2226:
2040:Indian mathematics
1993:
1922:
1867:
1709:
1672:of the trapezoid,
1632:
1562:
1477:and the diagonals
1420:
1335:
1239:
1227:
1204:
1193:
1183:height/altitude:
1172:
1151:
1123:
1095:
1069:
1026:
958:
911:
910:
709:
315:
280:Euclidean geometry
112:
107:
5153:Elementary shapes
5140:
5139:
4981:
4980:
4958:Myriagon (10,000)
4943:Triacontagon (30)
4907:Heptadecagon (17)
4897:Pentadecagon (15)
4892:Tetradecagon (14)
4831:Quadrilateral (4)
4701:Antiparallelogram
4549:"Right trapezoid"
4500:978-1-4020-6242-1
4092:978-1-4398-6489-0
3795:
3684:
3622:
3589:
3522:
3504:
3484:
3471:
3177:
3176:
3070:
3069:
2939:
2927:
2861:
2615:
2578:
2502:
2423:
2308:
2247:are parallel and
2224:
2211:
2147:
2107:
1976:
1865:
1840:
1704:
1627:
1330:
1320:
1310:
1196:{\displaystyle h}
1175:{\displaystyle m}
1052:parallel sides:
1040:Characterizations
639:
638:
618:No parallel sides
508:Non-parallelogram
347:Euclid's Elements
330:) 'four' + πέζα (
278:quadrilateral in
261:scalene trapezoid
135:
134:
106:
16:(Redirected from
5165:
4953:Chiliagon (1000)
4933:Icositrigon (23)
4912:Octadecagon (18)
4902:Hexadecagon (16)
4806:
4805:
4625:
4618:
4611:
4602:
4601:
4583:Trapezoidal Rule
4559:
4558:
4505:
4504:
4484:
4478:
4477:
4475:
4474:
4460:
4454:
4453:
4451:
4450:
4421:
4412:
4406:
4405:
4403:
4402:
4388:
4382:
4373:Deirdre Smeltzer
4369:
4360:
4359:
4357:
4356:
4342:
4336:
4327:
4321:
4314:
4285:
4275:
4269:
4263:
4257:
4256:
4255:
4238:
4215:
4214:
4212:
4211:
4200:
4194:
4193:
4191:
4190:
4180:
4174:
4173:
4171:
4170:
4156:
4150:
4149:
4138:
4132:
4131:
4103:
4097:
4096:
4076:
4067:
4052:
4046:
4045:
4034:
4028:
4027:
4013:
4002:
4001:
3990:
3984:
3977:
3971:
3970:
3968:
3967:
3953:
3824:Temple of Dendur
3809:
3807:
3806:
3801:
3796:
3791:
3790:
3752:
3746:
3698:
3696:
3695:
3690:
3685:
3683:
3669:
3655:
3640:
3638:
3637:
3632:
3627:
3623:
3621:
3610:
3596:
3590:
3582:
3549:Other properties
3541:
3539:
3538:
3533:
3528:
3524:
3523:
3521:
3510:
3505:
3503:
3492:
3485:
3477:
3472:
3470:
3459:
3372:
3368:
3366:
3365:
3360:
3347:
3343:
3341:
3340:
3335:
3322:
3318:
3316:
3315:
3310:
3297:
3293:
3291:
3290:
3285:
3272:
3268:
3266:
3265:
3260:
3247:
3243:
3241:
3240:
3235:
3188:
3186:
3185:
3180:
3178:
3175:
3164:
3163:
3162:
3147:
3146:
3128:
3127:
3115:
3114:
3101:
3100:
3084:
3082:
3081:
3076:
3071:
3068:
3057:
3056:
3055:
3040:
3039:
3021:
3020:
3008:
3007:
2994:
2993:
2953:
2951:
2950:
2945:
2940:
2938:
2937:
2932:
2928:
2923:
2922:
2921:
2909:
2908:
2896:
2895:
2883:
2882:
2872:
2862:
2860:
2859:
2858:
2833:
2829:
2828:
2813:
2812:
2794:
2793:
2781:
2780:
2762:
2761:
2746:
2745:
2727:
2726:
2714:
2713:
2697:
2695:
2654:
2652:
2651:
2646:
2617:
2608:
2589:
2587:
2586:
2581:
2579:
2505:
2503:
2501:
2500:
2486:
2480:
2469:
2437:
2435:
2434:
2429:
2424:
2311:
2309:
2307:
2306:
2292:
2283:
2272:
2235:
2233:
2232:
2227:
2225:
2223:
2222:
2217:
2213:
2212:
2210:
2199:
2198:
2197:
2185:
2184:
2174:
2148:
2140:
2135:
2134:
2125:
2108:
2100:
2044:Indian astronomy
2002:
2000:
1999:
1994:
1977:
1972:
1961:
1931:
1929:
1928:
1923:
1876:
1874:
1873:
1868:
1866:
1864:
1863:
1849:
1767:
1766:
1718:
1716:
1715:
1710:
1705:
1700:
1689:
1641:
1639:
1638:
1633:
1628:
1623:
1622:
1608:
1602:
1571:
1569:
1568:
1563:
1546:
1545:
1533:
1532:
1520:
1519:
1507:
1506:
1429:
1427:
1426:
1421:
1344:
1342:
1341:
1336:
1331:
1326:
1321:
1316:
1311:
1306:
1236:
1234:
1233:
1228:
1202:
1200:
1199:
1194:
1181:
1179:
1178:
1173:
1160:
1158:
1157:
1152:
1132:
1130:
1129:
1124:
1104:
1102:
1101:
1096:
1078:
1076:
1075:
1070:
1035:
1033:
1032:
1027:
1019:
1005:
997:
983:
967:
965:
964:
959:
920:
918:
917:
912:
894:
880:
872:
858:
777:point reflection
743:obtuse trapezoid
725:trapezoidal rule
659:proper trapezoid
605:
562:
549:Isosceles Trapez
541:Isosceles Trapez
521:
493:
478:ῥόμβος (rhombos)
475:
455:American English
435:(Definition 22)
406:
405:
231:
230:
227:
226:
223:
220:
217:
214:
211:
208:
205:
202:
199:
182:
181:
178:
177:
174:
171:
168:
165:
162:
159:
156:
153:
121:
119:
118:
113:
108:
102:
91:
46:
32:
31:
21:
5173:
5172:
5168:
5167:
5166:
5164:
5163:
5162:
5143:
5142:
5141:
5136:
5035:
4989:
4977:
4921:
4887:Tridecagon (13)
4877:Hendecagon (11)
4865:
4801:
4795:
4766:Right trapezoid
4687:
4639:
4629:
4529:
4514:
4512:Further reading
4509:
4508:
4501:
4485:
4481:
4472:
4470:
4462:
4461:
4457:
4448:
4446:
4436:10.2307/4145094
4430:(10): 853–863.
4419:
4413:
4409:
4400:
4398:
4390:
4389:
4385:
4370:
4363:
4354:
4352:
4344:
4343:
4339:
4328:
4324:
4315:
4288:
4276:
4272:
4264:
4260:
4239:
4218:
4209:
4207:
4201:
4197:
4188:
4186:
4182:
4181:
4177:
4168:
4166:
4158:
4157:
4153:
4140:
4139:
4135:
4126:
4104:
4100:
4093:
4077:
4070:
4053:
4049:
4036:
4035:
4031:
4014:
4005:
3992:
3991:
3987:
3978:
3974:
3965:
3963:
3955:
3954:
3950:
3945:
3920:
3908:
3869:
3857:
3840:
3816:
3786:
3748:
3747:
3745:
3734:
3731:
3730:
3670:
3656:
3654:
3652:
3649:
3648:
3611:
3597:
3595:
3591:
3581:
3573:
3570:
3569:
3551:
3514:
3509:
3496:
3491:
3490:
3486:
3476:
3463:
3458:
3456:
3453:
3452:
3424:is parallel to
3354:
3351:
3350:
3349:
3329:
3326:
3325:
3324:
3304:
3301:
3300:
3299:
3279:
3276:
3275:
3274:
3254:
3251:
3250:
3249:
3229:
3226:
3225:
3224:
3165:
3158:
3154:
3142:
3138:
3123:
3119:
3110:
3106:
3102:
3099:
3091:
3088:
3087:
3058:
3051:
3047:
3035:
3031:
3016:
3012:
3003:
2999:
2995:
2992:
2984:
2981:
2980:
2967:
2933:
2917:
2913:
2904:
2900:
2891:
2887:
2878:
2874:
2873:
2871:
2867:
2866:
2854:
2850:
2834:
2824:
2820:
2808:
2804:
2789:
2785:
2776:
2772:
2757:
2753:
2741:
2737:
2722:
2718:
2709:
2705:
2698:
2696:
2694:
2686:
2683:
2682:
2606:
2598:
2595:
2594:
2504:
2496:
2482:
2481:
2470:
2468:
2460:
2457:
2456:
2447:Heron's formula
2310:
2302:
2288:
2284:
2273:
2271:
2263:
2260:
2259:
2218:
2200:
2193:
2189:
2180:
2176:
2175:
2173:
2154:
2150:
2149:
2139:
2130:
2126:
2124:
2099:
2091:
2088:
2087:
2024:arithmetic mean
1962:
1960:
1952:
1949:
1948:
1938:
1893:
1890:
1889:
1859:
1845:
1841:
1765:
1757:
1754:
1753:
1690:
1688:
1680:
1677:
1676:
1650:
1618:
1604:
1603:
1601:
1593:
1590:
1589:
1541:
1537:
1528:
1524:
1515:
1511:
1502:
1498:
1496:
1493:
1492:
1376:
1373:
1372:
1325:
1315:
1305:
1303:
1300:
1299:
1215:
1212:
1211:
1188:
1185:
1184:
1182:
1167:
1164:
1163:
1161:
1139:
1136:
1135:
1133:
1111:
1108:
1107:
1105:
1084:
1081:
1080:
1057:
1054:
1053:
1051:
1042:
1015:
1001:
993:
979:
977:
974:
973:
968:, but it is an
929:
926:
925:
890:
876:
868:
854:
851:
848:
847:
817:
805:Euclidean plane
732:acute trapezoid
713:right trapezoid
701:
691:and non-square
644:
450:British English
307:
234:British English
196:
192:
150:
146:
92:
89:
87:
84:
83:
49:
37:Trapezium (BrE)
36:
35:Trapezoid (AmE)
28:
23:
22:
15:
12:
11:
5:
5171:
5161:
5160:
5155:
5138:
5137:
5135:
5134:
5129:
5124:
5119:
5114:
5109:
5104:
5099:
5094:
5092:Pseudotriangle
5089:
5084:
5079:
5074:
5069:
5064:
5059:
5054:
5049:
5043:
5041:
5037:
5036:
5034:
5033:
5028:
5023:
5018:
5013:
5008:
5003:
4998:
4992:
4990:
4983:
4982:
4979:
4978:
4976:
4975:
4970:
4965:
4960:
4955:
4950:
4945:
4940:
4935:
4929:
4927:
4923:
4922:
4920:
4919:
4914:
4909:
4904:
4899:
4894:
4889:
4884:
4882:Dodecagon (12)
4879:
4873:
4871:
4867:
4866:
4864:
4863:
4858:
4853:
4848:
4843:
4838:
4833:
4828:
4823:
4818:
4812:
4810:
4803:
4797:
4796:
4794:
4793:
4788:
4783:
4778:
4773:
4768:
4763:
4758:
4753:
4748:
4743:
4738:
4733:
4728:
4723:
4718:
4713:
4708:
4703:
4697:
4695:
4693:Quadrilaterals
4689:
4688:
4686:
4685:
4680:
4675:
4670:
4665:
4660:
4655:
4649:
4647:
4641:
4640:
4628:
4627:
4620:
4613:
4605:
4599:
4598:
4589:
4580:
4574:
4560:
4541:
4528:
4527:External links
4525:
4524:
4523:
4513:
4510:
4507:
4506:
4499:
4479:
4468:gogeometry.com
4455:
4407:
4396:www.efunda.com
4383:
4361:
4350:gogeometry.com
4337:
4322:
4286:
4270:
4258:
4216:
4195:
4175:
4151:
4133:
4098:
4091:
4068:
4047:
4029:
4003:
3985:
3972:
3947:
3946:
3944:
3941:
3940:
3939:
3933:
3927:
3919:
3916:
3907:
3904:
3879:outlined on a
3868:
3865:
3856:
3853:
3839:
3836:
3815:
3812:
3811:
3810:
3799:
3794:
3789:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3751:
3744:
3741:
3738:
3700:
3699:
3688:
3682:
3679:
3676:
3673:
3668:
3665:
3662:
3659:
3642:
3641:
3630:
3626:
3620:
3617:
3614:
3609:
3606:
3603:
3600:
3594:
3588:
3585:
3580:
3577:
3550:
3547:
3543:
3542:
3531:
3527:
3520:
3517:
3513:
3508:
3502:
3499:
3495:
3489:
3483:
3480:
3475:
3469:
3466:
3462:
3358:
3333:
3308:
3283:
3258:
3233:
3190:
3189:
3174:
3171:
3168:
3161:
3157:
3153:
3150:
3145:
3141:
3137:
3134:
3131:
3126:
3122:
3118:
3113:
3109:
3105:
3098:
3095:
3085:
3074:
3067:
3064:
3061:
3054:
3050:
3046:
3043:
3038:
3034:
3030:
3027:
3024:
3019:
3015:
3011:
3006:
3002:
2998:
2991:
2988:
2966:
2963:
2955:
2954:
2943:
2936:
2931:
2926:
2920:
2916:
2912:
2907:
2903:
2899:
2894:
2890:
2886:
2881:
2877:
2870:
2865:
2857:
2853:
2849:
2846:
2843:
2840:
2837:
2832:
2827:
2823:
2819:
2816:
2811:
2807:
2803:
2800:
2797:
2792:
2788:
2784:
2779:
2775:
2771:
2768:
2765:
2760:
2756:
2752:
2749:
2744:
2740:
2736:
2733:
2730:
2725:
2721:
2717:
2712:
2708:
2704:
2701:
2693:
2690:
2671:for a general
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2614:
2611:
2605:
2602:
2591:
2590:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2499:
2495:
2492:
2489:
2485:
2479:
2476:
2473:
2467:
2464:
2439:
2438:
2427:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2305:
2301:
2298:
2295:
2291:
2287:
2282:
2279:
2276:
2270:
2267:
2237:
2236:
2221:
2216:
2209:
2206:
2203:
2196:
2192:
2188:
2183:
2179:
2172:
2169:
2166:
2163:
2160:
2157:
2153:
2146:
2143:
2138:
2133:
2129:
2123:
2120:
2117:
2114:
2111:
2106:
2103:
2098:
2095:
2004:
2003:
1992:
1989:
1986:
1983:
1980:
1975:
1971:
1968:
1965:
1959:
1956:
1937:
1934:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1878:
1877:
1862:
1858:
1855:
1852:
1848:
1844:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1764:
1761:
1720:
1719:
1708:
1703:
1699:
1696:
1693:
1687:
1684:
1649:
1646:
1645:
1644:
1643:
1642:
1631:
1626:
1621:
1617:
1614:
1611:
1607:
1600:
1597:
1584:
1583:
1575:
1574:
1573:
1572:
1561:
1558:
1555:
1552:
1549:
1544:
1540:
1536:
1531:
1527:
1523:
1518:
1514:
1510:
1505:
1501:
1487:
1486:
1457:are parallel:
1447:
1446:
1443:
1440:
1437:
1430:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1365:
1357:
1356:
1348:
1347:
1346:
1345:
1334:
1329:
1324:
1319:
1314:
1309:
1294:
1293:
1282:
1279:
1272:
1265:
1258:
1226:
1222:
1219:
1192:
1171:
1150:
1146:
1143:
1122:
1118:
1115:
1094:
1091:
1088:
1068:
1064:
1061:
1041:
1038:
1025:
1022:
1018:
1014:
1011:
1008:
1004:
1000:
996:
992:
989:
986:
982:
957:
954:
951:
948:
945:
942:
939:
936:
933:
922:
921:
909:
906:
903:
900:
897:
893:
889:
886:
883:
879:
875:
871:
867:
864:
861:
857:
816:
813:
700:
697:
648:parallelograms
643:
640:
637:
636:
628:
619:
616:
608:τραπέζοειδὲς (
606:
599:
593:
592:
584:
576:
573:
563:
555:
554:
546:
538:
535:
525:
522:
515:
510:
504:
503:
500:
497:
494:
486:
485:
482:
479:
476:
469:
464:
458:
457:
452:
447:
442:
436:
430:
428:
426:
423:
422:
419:
416:
413:
410:
398:Charles Hutton
394:
393:
364:(τραπεζοειδή,
358:
306:
298:Etymology and
296:
133:
132:
127:
123:
122:
111:
105:
101:
98:
95:
81:
75:
74:
71:
61:
60:
55:
51:
50:
47:
39:
38:
26:
9:
6:
4:
3:
2:
5170:
5159:
5156:
5154:
5151:
5150:
5148:
5133:
5132:Weakly simple
5130:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5093:
5090:
5088:
5085:
5083:
5080:
5078:
5075:
5073:
5072:Infinite skew
5070:
5068:
5065:
5063:
5060:
5058:
5055:
5053:
5050:
5048:
5045:
5044:
5042:
5038:
5032:
5029:
5027:
5024:
5022:
5019:
5017:
5014:
5012:
5009:
5007:
5004:
5002:
4999:
4997:
4994:
4993:
4991:
4988:
4987:Star polygons
4984:
4974:
4973:Apeirogon (∞)
4971:
4969:
4966:
4964:
4961:
4959:
4956:
4954:
4951:
4949:
4946:
4944:
4941:
4939:
4936:
4934:
4931:
4930:
4928:
4924:
4918:
4917:Icosagon (20)
4915:
4913:
4910:
4908:
4905:
4903:
4900:
4898:
4895:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4874:
4872:
4868:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4834:
4832:
4829:
4827:
4824:
4822:
4819:
4817:
4814:
4813:
4811:
4807:
4804:
4798:
4792:
4789:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4759:
4757:
4754:
4752:
4751:Parallelogram
4749:
4747:
4746:Orthodiagonal
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4726:Ex-tangential
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4702:
4699:
4698:
4696:
4694:
4690:
4684:
4681:
4679:
4676:
4674:
4671:
4669:
4666:
4664:
4661:
4659:
4656:
4654:
4651:
4650:
4648:
4646:
4642:
4637:
4633:
4626:
4621:
4619:
4614:
4612:
4607:
4606:
4603:
4596:
4595:
4590:
4588:
4584:
4581:
4578:
4575:
4572:
4568:
4564:
4561:
4556:
4555:
4550:
4547:
4542:
4540:
4539:
4534:
4531:
4530:
4522:
4521:
4516:
4515:
4502:
4496:
4492:
4491:
4483:
4469:
4465:
4459:
4445:
4441:
4437:
4433:
4429:
4425:
4418:
4411:
4397:
4393:
4387:
4380:
4379:
4374:
4368:
4366:
4351:
4347:
4341:
4334:
4333:
4326:
4319:
4313:
4311:
4309:
4307:
4305:
4303:
4301:
4299:
4297:
4295:
4293:
4291:
4283:
4279:
4274:
4267:
4262:
4253:
4252:
4247:
4244:
4237:
4235:
4233:
4231:
4229:
4227:
4225:
4223:
4221:
4206:
4199:
4185:
4179:
4165:
4161:
4155:
4147:
4143:
4137:
4129:
4124:
4120:
4116:
4113:, Portuguese
4112:
4108:
4102:
4094:
4088:
4084:
4083:
4075:
4073:
4065:
4061:
4057:
4051:
4043:
4039:
4033:
4026:
4021:
4020:
4012:
4010:
4008:
3999:
3995:
3989:
3982:
3976:
3962:
3958:
3952:
3948:
3937:
3934:
3931:
3930:Polite number
3928:
3925:
3922:
3921:
3915:
3913:
3903:
3901:
3897:
3893:
3889:
3882:
3878:
3873:
3864:
3862:
3852:
3850:
3846:
3833:
3832:New York City
3829:
3825:
3820:
3797:
3792:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3742:
3739:
3736:
3729:
3728:
3727:
3725:
3722:intersect at
3721:
3717:
3713:
3710:intersect at
3709:
3705:
3686:
3680:
3677:
3674:
3671:
3666:
3663:
3660:
3657:
3647:
3646:
3645:
3628:
3624:
3618:
3615:
3612:
3607:
3604:
3601:
3598:
3592:
3586:
3583:
3578:
3575:
3568:
3567:
3566:
3564:
3560:
3556:
3546:
3529:
3525:
3518:
3515:
3511:
3506:
3500:
3497:
3493:
3487:
3481:
3478:
3473:
3467:
3464:
3460:
3451:
3450:
3449:
3447:
3443:
3439:
3438:harmonic mean
3435:
3431:
3427:
3423:
3419:
3415:
3411:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3379:
3374:
3371:
3346:
3321:
3296:
3271:
3246:
3222:
3218:
3214:
3209:
3207:
3203:
3199:
3195:
3172:
3169:
3166:
3159:
3155:
3151:
3148:
3143:
3139:
3135:
3132:
3129:
3124:
3120:
3116:
3111:
3107:
3103:
3096:
3093:
3086:
3072:
3065:
3062:
3059:
3052:
3048:
3044:
3041:
3036:
3032:
3028:
3025:
3022:
3017:
3013:
3009:
3004:
3000:
2996:
2989:
2986:
2979:
2978:
2977:
2971:
2962:
2960:
2941:
2934:
2929:
2924:
2918:
2914:
2910:
2905:
2901:
2897:
2892:
2888:
2884:
2879:
2875:
2868:
2863:
2855:
2847:
2844:
2841:
2835:
2825:
2821:
2817:
2814:
2809:
2805:
2801:
2798:
2795:
2790:
2786:
2782:
2777:
2773:
2769:
2758:
2754:
2750:
2747:
2742:
2738:
2734:
2731:
2728:
2723:
2719:
2715:
2710:
2706:
2702:
2691:
2688:
2681:
2680:
2679:
2676:
2674:
2673:quadrilateral
2670:
2666:
2662:
2658:
2657:semiperimeter
2639:
2636:
2633:
2630:
2627:
2624:
2621:
2612:
2609:
2603:
2600:
2572:
2569:
2566:
2563:
2560:
2551:
2548:
2545:
2542:
2539:
2530:
2527:
2524:
2515:
2512:
2509:
2493:
2490:
2487:
2477:
2474:
2471:
2465:
2462:
2455:
2454:
2453:
2450:
2448:
2444:
2425:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2299:
2296:
2293:
2285:
2280:
2277:
2274:
2268:
2265:
2258:
2257:
2256:
2254:
2250:
2246:
2242:
2219:
2214:
2207:
2204:
2201:
2194:
2190:
2186:
2181:
2177:
2170:
2164:
2161:
2158:
2151:
2144:
2141:
2136:
2131:
2127:
2118:
2115:
2112:
2104:
2101:
2096:
2093:
2086:
2085:
2084:
2082:
2078:
2074:
2070:
2066:
2061:
2059:
2055:
2051:
2050:
2045:
2041:
2037:
2033:
2032:mathematician
2029:
2025:
2021:
2017:
2013:
2009:
1990:
1987:
1984:
1981:
1978:
1973:
1969:
1966:
1963:
1957:
1954:
1947:
1946:
1945:
1943:
1933:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1887:
1883:
1856:
1853:
1850:
1842:
1834:
1831:
1828:
1825:
1822:
1813:
1810:
1807:
1804:
1801:
1792:
1789:
1786:
1777:
1774:
1771:
1762:
1759:
1752:
1751:
1750:
1748:
1744:
1740:
1736:
1735:perpendicular
1732:
1727:
1725:
1706:
1701:
1697:
1694:
1691:
1685:
1682:
1675:
1674:
1673:
1671:
1667:
1663:
1659:
1655:
1629:
1624:
1615:
1612:
1609:
1598:
1595:
1588:
1587:
1586:
1585:
1581:
1578:The distance
1577:
1576:
1559:
1556:
1553:
1550:
1547:
1542:
1538:
1534:
1529:
1525:
1521:
1516:
1512:
1508:
1503:
1499:
1491:
1490:
1489:
1488:
1484:
1480:
1476:
1472:
1468:
1464:
1460:
1459:
1458:
1456:
1452:
1444:
1441:
1438:
1435:
1431:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1370:
1366:
1363:
1359:
1358:
1354:
1350:
1349:
1332:
1327:
1322:
1317:
1312:
1307:
1298:
1297:
1296:
1295:
1291:
1287:
1283:
1280:
1277:
1273:
1270:
1266:
1263:
1259:
1256:
1252:
1251:supplementary
1248:
1244:
1243:
1242:
1224:
1220:
1217:
1208:
1190:
1169:
1162:midsegment:
1148:
1144:
1141:
1120:
1116:
1113:
1092:
1089:
1086:
1066:
1062:
1059:
1050:
1046:
1037:
1023:
1020:
1012:
1009:
1006:
998:
990:
987:
984:
971:
955:
952:
949:
946:
943:
940:
937:
934:
931:
907:
904:
901:
898:
895:
887:
884:
881:
873:
865:
862:
859:
846:
845:
844:
842:
838:
834:
830:
826:
822:
819:Four lengths
812:
810:
806:
802:
798:
793:
791:
787:
786:
780:
778:
774:
770:
769:
768:parallelogram
763:
761:
757:
756:
750:
748:
744:
739:
737:
733:
728:
726:
722:
718:
715:(also called
714:
705:
699:Special cases
696:
694:
690:
686:
681:
679:
675:
671:
666:
664:
660:
656:
651:
649:
635:
634:
629:
627:
626:
620:
617:
614:
613:
607:
604:
600:
598:
595:
594:
591:
590:
585:
583:
582:
577:
574:
571:
570:
564:
561:
557:
556:
553:
552:
547:
545:
544:
539:
536:
533:
532:
526:
520:
516:
514:
509:
505:
498:
492:
488:
487:
480:
474:
470:
468:
463:
462:Parallelogram
459:
456:
453:
451:
448:
446:
443:
440:
437:
434:
431:
429:
427:
425:
424:
414:
411:
408:
407:
404:
401:
399:
391:
387:
386:
381:
377:
376:
371:
367:
363:
359:
356:
352:
351:
350:
348:
344:
340:
337:Two types of
335:
333:
329:
325:
320:
311:
305:
301:
295:
293:
289:
285:
281:
277:
272:
270:
269:special cases
266:
262:
258:
257:lateral sides
254:
250:
245:
243:
239:
238:quadrilateral
235:
229:
190:
186:
180:
144:
140:
131:
128:
124:
109:
103:
99:
96:
93:
82:
80:
76:
72:
70:
66:
62:
59:
58:quadrilateral
56:
52:
45:
40:
33:
30:
19:
4926:>20 sides
4861:Decagon (10)
4846:Heptagon (7)
4836:Pentagon (5)
4826:Triangle (3)
4790:
4721:Equidiagonal
4592:
4586:
4552:
4536:
4519:
4489:
4482:
4471:. Retrieved
4467:
4458:
4447:. Retrieved
4427:
4423:
4410:
4399:. Retrieved
4395:
4386:
4376:
4353:. Retrieved
4349:
4340:
4330:
4325:
4278:Ask Dr. Math
4277:
4273:
4265:Trapezoids,
4261:
4249:
4208:. Retrieved
4198:
4187:. Retrieved
4178:
4167:. Retrieved
4163:
4154:
4145:
4136:
4122:
4118:
4114:
4110:
4106:
4101:
4081:
4055:
4050:
4041:
4032:
4023:
4018:
3997:
3988:
3980:
3975:
3964:. Retrieved
3960:
3951:
3912:multiplexors
3909:
3899:
3895:
3885:
3858:
3841:
3838:Architecture
3814:Applications
3723:
3719:
3715:
3711:
3707:
3703:
3701:
3643:
3562:
3558:
3552:
3544:
3445:
3441:
3433:
3429:
3425:
3421:
3417:
3413:
3409:
3405:
3401:
3397:
3393:
3389:
3385:
3381:
3377:
3375:
3369:
3344:
3319:
3294:
3269:
3244:
3220:
3216:
3212:
3210:
3205:
3201:
3197:
3193:
3191:
2975:
2956:
2677:
2592:
2451:
2442:
2440:
2252:
2248:
2244:
2240:
2238:
2080:
2076:
2072:
2068:
2062:
2054:special case
2047:
2019:
2015:
2011:
2007:
2005:
1941:
1939:
1885:
1881:
1879:
1746:
1742:
1738:
1730:
1728:
1721:
1669:
1665:
1661:
1653:
1651:
1579:
1482:
1478:
1474:
1470:
1466:
1462:
1454:
1450:
1448:
1368:
1352:
1289:
1285:
1240:
1134:diagonals:
1048:
923:
840:
836:
832:
828:
824:
820:
818:
794:
783:
781:
766:
764:
753:
751:
746:
742:
740:
735:
731:
729:
721:right angles
716:
712:
710:
682:
669:
667:
662:
658:
654:
652:
645:
632:
630:
624:
622:
611:
609:
596:
588:
586:
580:
578:
568:
566:
550:
548:
542:
540:
530:
528:
512:
507:
466:
461:
454:
449:
444:
438:
432:
402:
395:
383:
373:
365:
361:
354:
338:
336:
331:
327:
323:
316:
303:
299:
291:
287:
273:
260:
256:
252:
248:
246:
188:
142:
136:
29:
5122:Star-shaped
5097:Rectilinear
5067:Equilateral
5062:Equiangular
5026:Hendecagram
4870:11–20 sides
4851:Octagon (8)
4841:Hexagon (6)
4816:Monogon (1)
4658:Equilateral
4533:"Trapezium"
4246:"Trapezoid"
3900:trapeziform
3896:trapezoidal
3416:be on side
3408:be on side
2049:Aryabhatiya
382:-like' and
366:trapezoeidé
18:Trapezoidal
5147:Categories
5127:Tangential
5031:Dodecagram
4809:1–10 sides
4800:By number
4781:Tangential
4761:Right kite
4473:2018-02-13
4449:2016-04-06
4401:2024-05-15
4355:2024-05-15
4210:2023-06-09
4189:2008-04-14
4169:2007-12-10
4117:, Spanish
4109:, Italian
3966:2024-05-15
3943:References
3888:morphology
3881:spurge bug
3420:such that
2065:Bhāskara I
2036:astronomer
2030:, a great
1654:midsegment
1284:The areas
801:hyperbolic
693:rectangles
534:isoskelés)
286:cases. If
126:Properties
5107:Reinhardt
5016:Enneagram
5006:Heptagram
4996:Pentagram
4963:65537-gon
4821:Digon (2)
4791:Trapezoid
4756:Rectangle
4706:Bicentric
4668:Isosceles
4645:Triangles
4554:MathWorld
4251:MathWorld
4121:, German
3778:−
3769:−
3565:given by
3357:△
3332:△
3307:△
3282:△
3257:△
3232:△
3170:−
3133:−
3117:−
3063:−
3026:−
3010:−
2965:Diagonals
2911:−
2898:−
2864:−
2845:−
2799:−
2783:−
2732:−
2716:−
2570:−
2564:−
2549:−
2543:−
2528:−
2513:−
2491:−
2409:−
2403:−
2388:−
2376:−
2349:−
2316:−
2297:−
2205:−
2187:−
2162:−
2137:−
2028:Aryabhata
1979:⋅
1940:The area
1854:−
1832:−
1826:−
1811:−
1805:−
1790:−
1775:−
1724:bimedians
1658:midpoints
1613:−
1412:
1403:
1391:
1382:
1362:collinear
1276:triangles
1249:that are
1021:≠
1010:−
988:−
947:−
935:−
885:−
863:−
685:rhombuses
572:skalinón)
362:trapezoid
355:trapezium
304:trapezoid
300:trapezium
189:trapezium
143:trapezoid
5082:Isotoxal
5077:Isogonal
5021:Decagram
5011:Octagram
5001:Hexagram
4802:of sides
4731:Harmonic
4632:Polygons
4280:(2008),
4119:trapecio
4115:trapézio
4111:trapezio
3918:See also
3892:taxonomy
3877:pronotum
3855:Geometry
2959:bimedian
2058:triangle
1371:satisfy
1262:diagonal
790:incircle
674:calculus
670:at least
385:rhomboid
339:trapezia
324:trapezia
255:(or the
242:parallel
139:geometry
69:vertices
5102:Regular
5047:Concave
5040:Classes
4948:257-gon
4771:Rhombus
4711:Crossed
4535:at the
4444:4145094
4107:trapèze
4064:τράπεζα
3924:Frustum
3867:Biology
3826:in the
3726:, then
3436:is the
3432:. Then
2655:is the
2022:is the
1255:degrees
1106:legs:
689:squares
439:Proclus
392:-like')
390:rhombus
388:means '
378:means '
343:Proclus
302:versus
284:crossed
271:below.
244:sides.
236:, is a
5112:Simple
5057:Cyclic
5052:Convex
4776:Square
4716:Cyclic
4678:Obtuse
4673:Kepler
4597:(2008)
4497:
4442:
4123:Trapez
4089:
3555:lamina
3400:. Let
3388:, and
3192:where
2665:cyclic
2593:where
2239:where
2006:where
1880:where
1731:height
1351:where
1247:angles
1079:with
738:edge.
663:proper
631:Trapez
623:Trapez
610:trapez
587:Trapez
579:Trapez
567:trapez
529:trapez
433:Euclid
375:cuboid
328:tetrás
319:Euclid
276:convex
130:convex
5087:Magic
4683:Right
4663:Ideal
4653:Acute
4440:JSTOR
4420:(PDF)
3936:Wedge
2251:>
1269:ratio
612:oides
415:Image
409:Type
370:εἶδος
265:equal
249:bases
232:) in
187:, or
183:) in
65:Edges
5117:Skew
4741:Kite
4636:List
4495:ISBN
4087:ISBN
4060:πέζα
3859:The
3849:Inca
3822:The
3718:and
3706:and
3444:and
3428:and
3412:and
3396:and
3348:and
3298:and
3215:and
3204:and
2957:The
2243:and
2042:and
2010:and
1936:Area
1884:and
1729:The
1668:and
1652:The
1453:and
1369:ABCD
1288:and
1090:<
896:<
874:<
839:and
807:. A
775:(or
747:base
736:base
655:only
380:cube
332:péza
292:ABDC
288:ABCD
253:legs
141:, a
79:Area
67:and
54:Type
4585:on
4432:doi
4428:111
3898:or
3886:In
3830:in
3440:of
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730:An
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3468:G
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