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Trapezoid

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(Redirected fromTrapezoidal)
Convex quadrilateral with at least one pair of parallel sides

Trapezoid(American English)
Trapezium(British English)
Trapezoid or trapezium
Typequadrilateral
Edges andvertices4
Area12(a+b)h{\displaystyle {\tfrac {1}{2}}(a+b)h}
Propertiesconvex
Look up trapezoid in Wiktionary, the free dictionary.

Ingeometry, atrapezoid (/ˈtræpəzɔɪd/) inNorth American English, ortrapezium (/trəˈpiːziəm/) inBritish English,[1][2] is aquadrilateral that has at least one pair ofparallel sides.

The parallel sides are called thebases of the trapezoid.[3] The other two sides are called thelegs[3] orlateral sides. (If the trapezoid is aparallelogram, then the choice of bases and legs is arbitrary.)

A trapezoid is usually considered to be aconvex quadrilateral inEuclidean geometry, but there are alsocrossed cases. IfABCD is a convex trapezoid, thenABDC is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids.

Definitions

[edit]

Trapezoid can be defined exclusively or inclusively. Under an exclusive definition a trapezoid is a quadrilateral havingexactly one pair of parallel sides, with the other pair of opposite sides non-parallel. Parallelograms including rhombi, rectangles, and squares are then not considered to be trapezoids.[4][5] Under an inclusive definition, a trapezoid is any quadrilateral withat least one pair of parallel sides.[6] In an inclusive classification scheme, definitions are hierarchical: a square is a type of rectangle, a rectangle or rhombus is a type of parallelogram, and every parallelogram is a type of trapezoid.[7]

Professional mathematicians andpost-secondary geometry textbooks nearly always prefer inclusive definitions and classifications, because they simplify statements and proofs of geometric theorems.[8] In primary and secondary education, definitions ofrectangle andparallelogram are also nearly always inclusive, but an exclusive definition oftrapezoid is commonly found.[9][10] This article uses the inclusive definition and considers parallelograms to be special kinds of trapezoids. (Cf.Quadrilateral § Taxonomy.)

To avoid confusion, some sources use the termproper trapezoid to describe trapezoids with exactly one pair of parallel sides, analogous to uses of the wordproper in some other mathematical objects.[11][12]

Etymology

[edit]

In the ancient Greek geometry ofEuclid'sElements (c. 300 BC), quadrilaterals were classified into exclusive categories:square; oblong (non-squarerectangle); (non-square)rhombus; rhomboid, meaning a non-rhombus non-rectangleparallelogram; or trapezium (τραπέζιον, literally "table"), meaning any quadrilateral not already included in the previous categories.[13]

The Neoplatonist philosopherProclus (mid 5th century AD) wrote an influential commentary on Euclid with a richer set of categories, which he attributed toPosidonius (c. 100 BC). In this scheme, a quadrilateral can be a parallelogram or a non-parallelogram. A parallelogram can itself be a square, an oblong (non-square rectangle), a rhombus, or a rhomboid (non-rhombus non-rectangle). A non-parallelogram can be atrapezium with exactly one pair of parallel sides, which can beisosceles (with equal legs) orscalene (with unequal legs); or atrapezoid (τραπεζοειδή, literally "table-like") with no parallel sides.[13][14]

Hutton's definitions in 1795

All European languages except for English follow Proclus's meanings oftrapezium andtrapezoid,[15] as did English until the late 18th century, when an influential mathematical dictionary published byCharles Hutton in 1795 transposed the two terms without explanation, leading to widespread inconsistency. Hutton's change was reversed in British English in about 1875, but it has been retained in American English to the present.[13] Late 19th century American geometry textbooks define a trapezium as havingno parallel sides, a trapezoid as havingexactly one pair of parallel sides, and a parallelogram as having two sets of opposing parallel sides.[3][16]To avoid confusion between contradictory British and American meanings oftrapezium andtrapezoid, quadrilaterals with no parallel sides have sometimes been calledirregular quadrilaterals.[17]

Special cases

[edit]
Trapezoid special cases. The orange figures also qualify as parallelograms.

Anisosceles trapezoid is a trapezoid where the base angles have the same measure.[18][19] As a consequence the two legs are also of equal length and it hasreflection symmetry.[20] This is possible for acute trapezoids or right trapezoids as rectangles. An acute trapezoid is a trapezoid with two adjacent acute angles on its longer base, and the isosceles trapezoid is an example of an acute trapezoid. The isosceles trapezoid has a special case known as a three-sided trapezoid, meaning it is a trapezoid wherein two trapezoid's legs have equal lengths as the trapezoid's base at the top.[21] The isosceles trapezoid is theconvex hull of anantiparallelogram, a type ofcrossed quadrilateral. Every antiparallelogram is formed with such a trapezoid by replacing two parallel sides by the two diagonals.[22]

An obtuse trapezoid, on the other hand, has one acute and one obtuse angle on each base. An example isparallelogram with equal acute angles.[21]

A right trapezoid is a trapezoid with two adjacentright angle. One special type of right trapezoid is by forming threeright triangles,[23] which was used byJames Garfield toprove thePythagorean theorem.[24]

Atangential trapezoid is a trapezoid that has anincircle.

Condition of existence

[edit]

Four lengthsa,c,b,d can constitute the consecutive sides of a non-parallelogram trapezoid witha andb parallel only when[25]

|d−c|<|b−a|<d+c.{\displaystyle \displaystyle |d-c|<|b-a|<d+c.}

The quadrilateral is a parallelogram whend−c=b−a=0{\displaystyle d-c=b-a=0}, but it is anex-tangential quadrilateral (which is not a trapezoid) when|d−c|=|b−a|≠0{\displaystyle |d-c|=|b-a|\neq 0}.[26]

Characterizations

[edit]
general trapezoid/trapezium:
parallel sides:a,b{\displaystyle a,\,b} witha<b{\displaystyle a<b}
legs:c,d{\displaystyle c,\,d}
diagonals:q,p{\displaystyle q,\,p}
midsegment:m{\displaystyle m}
height/altitude:h{\displaystyle h}
trapezoid/trapezium with opposing trianglesS,T{\displaystyle S,\,T} formed by the diagonals

Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:

  • It has two adjacentangles that aresupplementary, that is, they add up to 180degrees.
  • The angle between a side and adiagonal is equal to the angle between the opposite side and the same diagonal.
  • The diagonals cut each other in mutually the sameratio (this ratio is the same as that between the lengths of the parallel sides).
  • The diagonals cut the quadrilateral into fourtriangles of which one opposite pair have equal areas.[27]
  • 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.[28]
  • The areasS andT of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation
K=S+T,{\displaystyle {\sqrt {K}}={\sqrt {S}}+{\sqrt {T}},}
whereK is the area of the quadrilateral.[29]
  • The midpoints of two opposite sides of the trapezoid and the intersection of the diagonals arecollinear.[30]
  • The angles in the quadrilateralABCD satisfysin⁡Asin⁡C=sin⁡Bsin⁡D.{\displaystyle \sin A\sin C=\sin B\sin D.}[31]
  • The cosines of two adjacent anglessum to 0, as do the cosines of the other two angles.[31]
  • The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.[32]
  • One bimedian divides the quadrilateral into two quadrilaterals of equal areas.[32]
  • Twice the length of the bimedian connecting the midpoints of two opposite sides equals the sum of the lengths of the other sides.[33]

Additionally, the following properties are equivalent, and each implies that opposite sidesa andb are parallel:

  • The consecutive sidesa,c,b,d and the diagonalsp,q satisfy the equation[34]
p2+q2=c2+d2+2ab.{\displaystyle p^{2}+q^{2}=c^{2}+d^{2}+2ab.}
  • The distancev between the midpoints of the diagonals satisfies the equation[35]
v=|a−b|2.{\displaystyle v={\frac {|a-b|}{2}}.}

Properties

[edit]

Midsegment and height

[edit]

Themidsegment ormedian of a trapezoid is the segment that joins themidpoints of the legs. It is parallel to the bases. Its lengthm is equal to the average of the lengths of the basesa andb of the trapezoid,[36][19][37][6]

m=a+b2.{\displaystyle m={\frac {a+b}{2}}.}

The midsegment of a trapezoid is one of the twobimedians (the other bimedian divides the trapezoid into equal areas).

Theheight (or altitude) is theperpendicular distance between the bases.[3] In the case that the two bases have different lengths (a ≠b), the height of a trapezoidh can be determined by the length of its four sides using the formula[38]

h=(p−a)(p−b)(p−b−d)(p−b−c)2|b−a|{\displaystyle h={\frac {\sqrt {(p-a)(p-b)(p-b-d)(p-b-c)}}{2|b-a|}}}

wherec andd are the lengths of the legs andp=a+b+c+d{\displaystyle p=a+b+c+d}.

Area

[edit]

The areaK{\displaystyle K} of a trapezoid is given by the product of the midsegment (the average of the two bases) and the height:K=12(a+b)h{\displaystyle K={\tfrac {1}{2}}(a+b)h}wherea{\displaystyle a} andb{\displaystyle b} are the lengths of the bases, andh{\displaystyle h} is the height (the perpendicular distance between these sides).[39] This method has been used inAryabhata'sAryabhatiya in section 2.8 in the classical age of Indian, yielding as aspecial case the well-known formula for the area of atriangle, by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.

The 7th-century Indian mathematicianBhāskara I derived the following formula for the area of a trapezoid with consecutive sidesa{\displaystyle a},c{\displaystyle c},b{\displaystyle b},d{\displaystyle d}::K=12(a+b)c2−14((b−a)+c2−d2b−a)2{\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}}}}wherea{\displaystyle a} andb{\displaystyle b} are parallel andb>a{\displaystyle b>a}.[40] This formula can be factored into a more symmetric version[38]

K=a+b4|b−a|(−a+b+c+d)(a−b+c+d)(a−b+c−d)(a−b−c+d).{\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)}}.}

When one of the parallel sides has shrunk to a point (saya = 0), this formula reduces toHeron's formula for the area of a triangle.

Another equivalent formula for the area, which more closely resembles Heron's formula, is[38]

K=a+b|b−a|(s−b)(s−a)(s−b−c)(s−b−d){\displaystyle K={\frac {a+b}{|b-a|}}{\sqrt {(s-b)(s-a)(s-b-c)(s-b-d)}}}

wheres=12(a+b+c+d){\displaystyle s={\tfrac {1}{2}}(a+b+c+d)} is thesemiperimeter of the trapezoid. (This formula is similar toBrahmagupta's formula, but it differs from it, in that a trapezoid might not becyclic (inscribed in a circle). The formula is also a special case ofBretschneider's formula for a generalquadrilateral).

From Bretschneider's formula, it follows that

K=(ab2−a2b−ad2+bc2)(ab2−a2b−ac2+bd2)4(b−a)2−(c2+d2−a2−b24)2.{\displaystyle K={\sqrt {{\frac {\left(ab^{2}-a^{2}b-ad^{2}+bc^{2}\right)\left(ab^{2}-a^{2}b-ac^{2}+bd^{2}\right)}{4(b-a)^{2}}}-\left({\frac {c^{2}+d^{2}-a^{2}-b^{2}}{4}}\right)^{2}}}.}

Thebimedian connecting the parallel sides bisects the area. More generally, any line drawn through the midpoint of the median parallel to the bases, that intersects the bases, bisects the area. Any triangle connecting the two ends of one leg to the midpoint of the other leg is also half of the area.[41]

Diagonals

[edit]

The lengths of the diagonals arep=ab2−a2b−ac2+bd2b−a,q=ab2−a2b−ad2+bc2b−a,{\displaystyle {\begin{aligned}p&={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},\\q&={\sqrt {\frac {ab^{2}-a^{2}b-ad^{2}+bc^{2}}{b-a}}},\end{aligned}}}wherea{\displaystyle a} is the short base,b{\displaystyle b} is the long base, andc{\displaystyle c} andd{\displaystyle d} are the trapezoid legs.[42]

If the trapezoid is divided into four triangles by its diagonalsAC andBD (as shown on the right), intersecting atO, then the area of△{\displaystyle \triangle }AOD is equal to that of△{\displaystyle \triangle }BOC, and the product of the areas of△{\displaystyle \triangle }AOD and△{\displaystyle \triangle }BOC is equal to that of△{\displaystyle \triangle }AOB and△{\displaystyle \triangle }COD. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.[38]

If⁠ℓ{\displaystyle \ell }⁠ is the length of the line segment parallel to the bases, passing through the intersection of the diagonals, with one endpoint on each leg, then⁠ℓ{\displaystyle \ell }⁠ is theharmonic mean of the lengths of the bases:[43]

1ℓ=12(1a+1b).{\displaystyle {\frac {1}{\ell }}={\frac {1}{2}}\left({\frac {1}{a}}+{\frac {1}{b}}\right).}

The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.[44]

Other properties

[edit]

The center of area (center of mass for a uniformlamina) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distancex from the longer sideb given by[45]

x=h3(2a+ba+b).{\displaystyle x={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).}

The center of area divides this segment in the ratio (when taken from the short to the long side)[46]: p. 862 

a+2b2a+b.{\displaystyle {\frac {a+2b}{2a+b}}.}

If the angle bisectors to anglesA andB intersect atP, and the angle bisectors to anglesC andD intersect atQ, then[44]

PQ=|AD+BC−AB−CD|2.{\displaystyle PQ={\frac {|AD+BC-AB-CD|}{2}}.}

Applications

[edit]
The trapezoidal rule for numerical integration
Example of a trapeziformpronotum outlined on aspurge bug

Incalculus, thedefinite integral of afunctionf(x){\displaystyle f(x)} can benumerically approximated as a discretesum bypartitioning the interval of integration into small uniform intervals and approximating the function's value on each interval as the average of the values at its endpoints:∫abf(x)dx≈∑k=1N12(f(xk−1)+f(xk))Δx,{\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\tfrac {1}{2}}{\bigl (}f(x_{k-1})+f(x_{k}){\bigr )}\Delta x,}whereN{\displaystyle N} is the number of intervals,x0=a{\displaystyle x_{0}=a},xN=b{\displaystyle x_{N}=b}, andΔx=(b−a)/N{\displaystyle \Delta x=(b-a)/N}. Graphically, this amounts to approximating the region under thegraph of the function by a collection of trapezoids, so this method is called thetrapezoidal rule.[47]

When any rectangle is viewed inperspective from a position which is centered on one axis but not the other, it appears to be an isosceles trapezoid, called thekeystone effect because archkeystones are commonly trapezoidal. For example, when a rectangular buildingfaçade is photographed from the ground at a position directly in front using arectilinear lens, the image of the building is an isosceles trapezoid. Such photographs sometimes have a "keystone transformation" applied to them to recover rectangular shapes.Video projectors sometimes apply such a keystone transformation to the recorded image before projection, so that the image projected on a flat screen appears undistorted.

Piazza del Campidoglio viewed from directly above.

Trapezoids, especially isosceles trapezoids, are common in architecture, in the shapes of doors, windows, and whole buildings.[citation needed] This was the standard style for the doors and windows of theInca, although it can be found used by earlier cultures of the same region and did not necessarily originate with them.[48][49]An almena, abattlement feature characteristic ofMoorish architecture, is trapezoidal.[50]Michaelangelo's redesign of thePiazza del Campidoglio (see photograph at right) incorporated a trapezoid surrounding an ellipse, giving the effect of a square surrounding a circle when seen foreshortened at ground level.[51]Cinematography takes advantage of trapezoids in the opposite way, to produce an excessive foreshortening effect from the camera viewpoint, giving the illusion of greater depth to a room in a movie studio than the set physically has.[52]Trapezoids were also used to produce the visual distortions ofCaligarism.[52]Canals and drainageditches commonly have a trapezoidal cross-section.

In biology, especiallymorphology andtaxonomy, terms such astrapezoidal ortrapeziform commonly are useful in descriptions of particular organs or forms.[53]

Trapezoids are sometimes used as a graphical symbol. Incircuit diagrams, a trapezoid is the symbol for amultiplexer.[54] An isosceles trapezoid is used for the shape of road signs, for example, on secondary highways inOntario, Canada.[55]

Non-Euclidean geometry

[edit]

Inspherical orhyperbolic geometry, the internal angles of a quadrilateral do not sum to 360°, but quadrilaterals analogous to trapezoids, parallelograms, and rectangles can still be defined, and additionally there are a few new types of quadrilaterals not distinguished in the Euclidean case.

A spherical or hyperbolic trapezoid is a quadrilateral with two opposite sides, the legs, each of whose two adjacent angles sum to the same quantity; the other two sides are the bases.[56] As in Euclidean geometry, special cases include isosceles trapezoids whose legs are equal (as are the angles adjacent to each base), parallelograms with two pairs of opposite equal angles and two pairs of opposite equal sides, rhombuses with two pairs of opposite equal angles and four equal sides, rectangles with four equal (non-right) angles and two pairs of opposite equal sides, and squares with four equal (non-right) angles and four equal sides.

When a rectangle is cut in half along the line through the midpoints of two opposite sides, each of the resulting two pieces is an isosceles trapezoid with two right angles, called aSaccheri quadrilateral. When a rectangle is cut into quarters by the two lines through pairs of opposite midpoints, each of the resulting four pieces is a quadrilateral with three right angles called aLambert quadrilateral. In Euclidean geometry Saccheri and Lambert quadrilaterals are merely rectangles.

Related topics

[edit]
An example of trapezoidal number: 15 = 4 + 5 + 6

Thetrapezoidal number is a set ofpositive integer obtained by summing consecutively two or more positive integers greater than one, forming a trapezoidal pattern.[57]

Thecrossed ladders problem 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.[58]

See also

[edit]
  • Frustum, a solid having trapezoidal faces
  • Wedge, a polyhedron defined by two triangles and three trapezoid faces.

Notes

[edit]
  1. ^"Trapezoid – math word definition – Math Open Reference".www.mathopenref.com. Retrieved2024-05-15.
  2. ^Gardiner, Anthony D.; Bradley, Christopher J. (2005).Plane Euclidean Geometry: Theory and Problems. United Kingdom Mathematics Trust. p. 34.ISBN 9780953682362.
  3. ^abcdHopkins 1891, p. 33.
  4. ^Usiskin & Griffin 2008, p. 29.
  5. ^Alsina & Nelsen 2020, p. 90.
  6. ^abRingenberg, Lawrence A. (1977). "Coordinates in a Plane".College Geometry. R. E. Krieger Publishing Company. pp. 161–162.ISBN 9780882755458.
  7. ^Alsina & Nelsen 2020, p. 89.
  8. ^Usiskin & Griffin 2008, p. 32.
  9. ^Craine, Timothy V.; Rubenstein, Rheta N. (1993). "A Quadrilateral Hierarchy to Facilitate Learning in Geometry".The Mathematics Teacher.86 (1):30–36.doi:10.5951/MT.86.1.0030.JSTOR 27968085.
  10. ^Popovic, Gorjana (2012). "Who is This Trapezoid, Anyway?".Mathematics Teaching in the Middle School.18 (4):196–199.doi:10.5951/mathteacmiddscho.18.4.0196.JSTOR 10.5951/mathteacmiddscho.18.4.0196.ResearchGate:259750174.
  11. ^Michon, Gérard P."History and Nomenclature". Retrieved2023-06-09.
  12. ^Beem, John K. (2006).Geometry Connections: Mathematics for Middle School Teachers. Connections in mathematics courses for teachers. Pearson Prentice Hall. p. 57.ISBN 9780131449268.
  13. ^abcMurray, James (1926)."Trapezium".A New English Dictionary on Historical Principles: Founded Mainly on the Materials Collected by the Philological Society. Vol. X. Clarendon Press at Oxford. p. 286, also see "Trapezoid", pp. 286–287.
  14. ^Morrow, Glenn R., ed. (1970).Proclus: A commentary on the first book of Euclid's Elements. Princeton University Press.§§ 169–174, pp. 133–137.
  15. ^Conway, Burgiel & Goodman-Strauss 2016, p. 286.
  16. ^Hobbs 1899, p. 66.
  17. ^Davies, Charles (1873).The Nature and Utility of Mathematics. New York: A.S. Barnes & Company. p. 35.
  18. ^Dodge 2012, p. 82.
  19. ^abPosamentier, Alfred S.; Bannister, Robert L. (2014). "The Trapezoid".Geometry, Its Elements and Structure: Second Edition. Dover Books on Mathematics (2nd ed.). Courier Corporation. §7.7, pp. 282–287.ISBN 9780486782164.
  20. ^Hopkins 1891, p. 34.
  21. ^abAlsina & Nelsen 2020, p. 90–91.
  22. ^Alsina & Nelsen 2020, p. 212.
  23. ^Alsina & Nelsen 2020, p. 91.
  24. ^Garfield, James (1876). "Pons Asinorum".New England Journal of Education.3 (14): 161.ISSN 2578-4145.JSTOR 44764657.
  25. ^Ask Dr. Math (2008),"Area of Trapezoid Given Only the Side Lengths".
  26. ^Josefsson 2013, p. 35.
  27. ^Josefsson 2013, Prop. 5.
  28. ^Josefsson 2013, Thm. 6.
  29. ^Josefsson 2013, Thm. 8.
  30. ^Josefsson 2013, Thm. 15.
  31. ^abJosefsson 2013, p. 25.
  32. ^abJosefsson 2013, p. 26.
  33. ^Josefsson 2013, p. 31.
  34. ^Josefsson 2013, Cor. 11.
  35. ^Josefsson 2013, Thm. 12.
  36. ^Hobbs 1899, p. 58.
  37. ^Dodge 2012, p. 117.
  38. ^abcdWeisstein, Eric W."Trapezoid".MathWorld.
  39. ^Dodge 2012, p. 84.
  40. ^Puttaswamy, T. K. (2012).Mathematical Achievements of Pre-modern Indian Mathematicians. Elsevier. p. 156.ISBN 978-0-12-397913-1.
  41. ^Hopkins 1891, p. 95.
  42. ^Alsina & Nelsen 2020, p. 96.
  43. ^Skidell, Akiva (1977). "The Harmonic Mean: A Nomograph, and some Problems".The Mathematics Teacher.70 (1):30–34.doi:10.5951/MT.70.1.0030.JSTOR 27960699.
    Hoehn, Larry (1984). "A Geometrical Interpretation of the Weighted".Two-Year College Mathematics Journal.15 (2):135–139.doi:10.1080/00494925.1984.11972762.
  44. ^abOwen Byer, Felix Lazebnik andDeirdre Smeltzer,Methods for Euclidean Geometry, Mathematical Association of America, 2010, p. 55.
  45. ^"Centroid, Area, Moments of Inertia, Polar Moments of Inertia, & Radius of Gyration of a General Trapezoid".www.efunda.com. Retrieved2024-05-15.
  46. ^Apostol, Tom M.; Mnatsakanian, Mamikon A. (December 2004)."Figures Circumscribing Circles"(PDF).American Mathematical Monthly.111 (10):853–863.doi:10.2307/4145094.JSTOR 4145094. Retrieved2016-04-06.
  47. ^Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007).Calculus (9th ed.).Pearson Prentice Hall. p. 264.ISBN 978-0131469686.
  48. ^"Machu Picchu Lost City of the Incas – Inca Geometry".gogeometry.com. Retrieved2018-02-13.
  49. ^Hyslop, John (2014).Inka Settlement Planning. University of Texas Press. p. 54.ISBN 9780292762640.
  50. ^Curl 1999, p. 19, almena.
  51. ^Curl 1999, p. 486, Michaelangelo Buonarroti.
  52. ^abRamírez 2012, p. 84.
  53. ^John L. Capinera (11 August 2008).Encyclopedia of Entomology. Springer Science & Business Media. pp. 386, 1062, 1247.ISBN 978-1-4020-6242-1.
  54. ^Daniels, Jerry (1996).Digital Design from Zero to One. John Wiley & Sons.p. 203.
  55. ^Alsina & Nelsen 2020, p. 93.
  56. ^Petrov, F. V. (2009).Вписанные четырёхугольники и трапеции в абсолютной геометрии [Cyclic quadrilaterals and trapezoids in absolute geometry](PDF).Matematicheskoe Prosveschenie. Tret’ya Seriya (in Russian).13:149–154.
  57. ^Gamer, Carlton; Roeder, David W.; Watkins, John J. (1985). "Trapezoidal numbers".Mathematics Magazine.58 (2):108–110.doi:10.2307/2689901.JSTOR 2689901.
  58. ^Alsina & Nelsen (2020), p. 102.

Bibliography

[edit]
  • Alsina, Claudi; Nelsen, Roger (2020).A Cornucopia of Quadrilaterals. Mathematical Association of America.ISBN 978-1-4704-5312-1.
  • Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2016).The Symmetries of Things. CRC Press.ISBN 978-1-4398-6489-0.
  • Curl, James Stevens (1999).A Dictionary of Architecture. Oxford University Press.ISBN 9780198606789.
  • Dodge, Clayton W. (2012).Euclidean Geometry and Transformations. Dover Books on Mathematics. Courier Corporation.ISBN 9780486138428.
  • Hobbs, Charles Austin (1899).The Elements of Plane Geometry. A. Lovell & Company.
  • Hopkins, George Irving (1891).Manual of Plane Geometry. D.C. Heath & Company.
  • Josefsson, Martin (2013)."Characterizations of trapezoids"(PDF).Forum Geometricorum.13:23–35. Archived fromthe original(PDF) on 16 June 2013.
  • Usiskin, Zalman; Griffin, Jennifer (2008).The Classification of Quadrilaterals: A Study of Definition.Information Age Publishing. pp. 49–52,63–67.
  • Ramírez, Juan Antonio (2012). "Architecture and Desire: The character of film constructions".Architecture for the Screen: A Critical Study of Set Design in Hollywood's Golden Age. Translated by Moffitt, John F. McFarland.ISBN 9780786469307.

Further reading

[edit]

External links

[edit]
Triangles
Quadrilaterals
By number
of sides
1–10 sides
11–20 sides
>20 sides
Star polygons
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