how to draw a 3d trapezoid

Trapezoid (AmE) Trapezium (BrE)
Trapezoid
Type	quadrilateral
Edges and vertices	4
Area	${\displaystyle {\tfrac {a+b}{ii}}h}$
Properties	convex

In Euclidean geometry, a convex quadrilateral with at least 1 pair of parallel sides is referred to equally a trapezoid ^{[1] [2]} () in American and Canadian English simply every bit a trapezium () in English outside North America. The parallel sides are called the bases of the trapezoid and the other two sides are chosen the legs or the lateral sides (if they are not parallel; otherwise at that place are ii pairs of bases). A scalene trapezoid is a trapezoid with no sides of equal measure,^[three] in contrast to the special cases beneath.

Contents

• ane Etymology
• ii Inclusive vs exclusive definition
• 3 Special cases
• 4 Condition of being
• 5 Characterizations
• 6 Midsegment and height
• 7 Expanse
• eight Diagonals
• nine Other properties
• 10 More than on terminology
• xi Applications
  • eleven.i Architecture
  • 11.2 Geometry
  • eleven.3 Biological science
• 12 See too
• 14 External links

Etymology

The term trapezium has been in apply in English since 1570, from Late Latin trapezium, from Greek τραπέζιον (trapézion), literally "a piddling table", a diminutive of τράπεζα (trápeza), "a tabular array", itself from τετράς (tetrás), "four" + πέζα (péza), "a foot, an edge". The first recorded use of the Greek word translated trapezoid (τραπεζοειδή, trapezoeidé, "tabular array-like") was past Marinus Proclus (412 to 485 AD) in his Commentary on the first book of Euclid'southward Elements.^[iv]

This commodity uses the term trapezoid in the sense that is electric current in the United States and Canada. In many other languages using a give-and-take derived from the Greek for this figure, the form closest to trapezium (e.g. Portuguese trapézio, French trapèze, Italian trapezio, Castilian trapecio, German language Trapez, Russian "трапеция") is used.

Inclusive vs exclusive definition

There is some disagreement whether parallelograms, which have two pairs of parallel sides, should exist regarded every bit trapezoids. Some ascertain a trapezoid equally a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms.^[5] Others^{[half dozen]} define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition^[7]), making the parallelogram a special blazon of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. The sometime definition would brand such concepts as the trapezoidal approximation to a definite integral ill-defined. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is as well advocated in the taxonomy of quadrilaterals.

Under the inclusive definition, all parallelograms (including rhombuses, rectangles and squares) are trapezoids. Rectangles have mirror symmetry on mid-edges; rhombuses take mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices.

Special cases

A correct trapezoid (likewise called right-angled trapezoid) has two next right angles.^[6] Right trapezoids are used in the trapezoidal rule for estimating areas under a bend.

An acute trapezoid has two next acute angles on its longer base edge, while an obtuse trapezoid has one astute and 1 obtuse angle on each base.

An acute trapezoid is likewise an isosceles trapezoid, if its sides (legs) have the same length, and the base angles have the same mensurate. It has reflection symmetry.

An obtuse trapezoid with 2 pairs of parallel sides is a parallelogram. A parallelogram has central 2-fold rotational symmetry (or point reflection symmetry).

A Saccheri quadrilateral is similar to a trapezoid in the hyperbolic airplane, with two adjacent correct angles, while information technology is a rectangle in the Euclidean aeroplane. A Lambert quadrilateral in the hyperbolic airplane has 3 right angles.

A tangential trapezoid is a trapezoid that has an incircle.

Condition of existence

Four lengths a, c, b, d can found the consecutive sides of a not-parallelogram trapezoid with a and b parallel only when^[viii]

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

The quadrilateral is a parallelogram when ${\displaystyle d-c=b-a=0}$ , but it is an ex-tangential quadrilateral (which is non a trapezoid) when ${\displaystyle |d-c|=|b-a|\neq 0}$ .^{[9] :p. 35}

Characterizations

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

• It has two adjacent angles that are supplementary, that is, they add upwards to 180 degrees.
• The angle between a side and a diagonal is equal to the angle between the reverse side and the same diagonal.
• The diagonals cut each other in mutually the same ratio (this ratio is the aforementioned as that between the lengths of the parallel sides).
• The diagonals cut the quadrilateral into four triangles of which ane opposite pair are similar.
• The diagonals cut the quadrilateral into iv triangles of which ane opposite pair have equal areas.^{[9] :Prop.v}
• 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.^{[9] :Thm.6}
• The areas S and T of some two opposite triangles of the iv triangles formed past the diagonals satisfy the equation

${\displaystyle {\sqrt {Thousand}}={\sqrt {South}}+{\sqrt {T}},}$

where K is the area of the quadrilateral.^{[9] :Thm.viii}

• The midpoints of two opposite sides and the intersection of the diagonals are collinear.^{[9] :Thm.xv}
• The angles in the quadrilateral ABCD satisfy ${\displaystyle \sin A\sin C=\sin B\sin D.}$ ^{[9] :p. 25}
• The cosines of two adjacent angles sum to 0, as practise the cosines of the other 2 angles.^{[nine] :p. 25}
• The cotangents of two side by side angles sum to 0, as do the cotangents of the other ii side by side angles.^{[9] :p. 26}
• One bimedian divides the quadrilateral into ii quadrilaterals of equal areas.^{[9] :p. 26}
• Twice the length of the bimedian connecting the midpoints of two contrary sides equals the sum of the lengths of the other sides.^{[ix] :p. 31}

Additionally, the following properties are equivalent, and each implies that opposite sides a and b are parallel:

• The consecutive sides a, c, b, d and the diagonals p, q satisfy the equation^{[9] :Cor.xi}

${\displaystyle p^{ii}+q^{2}=c^{ii}+d^{2}+2ab.}$

• The distance v between the midpoints of the diagonals satisfies the equation^{[nine] :Thm.12}

${\displaystyle v={\frac {|a-b|}{2}}.}$

Midsegment and peak

The midsegment (too called the median or midline) of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases. Its length grand is equal to the average of the lengths of the bases a and b of the trapezoid,^[6]

${\displaystyle one thousand={\frac {a+b}{ii}}.}$

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

The elevation (or altitude) is the perpendicular distance betwixt the bases. In the instance that the 2 bases have different lengths (a ≠ b), the height of a trapezoid h tin be determined past the length of its iv sides using the formula^[six]

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

where c and d are the lengths of the legs.

Area

The area K of a trapezoid is given past^{[half dozen]}

${\displaystyle K={\frac {a+b}{2}}\cdot h=mh}$

where a and b are the lengths of the parallel sides, h is the height (the perpendicular distance between these sides), and m is the arithmetic mean of the lengths of the two parallel sides. In 499 Advert Aryabhata, a nifty mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the Aryabhatiya (section 2.8). This yields every bit a special case the well-known formula for the area of a triangle, by because a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.

The 7th-century Indian mathematician Bhāskara I derived the following formula for the surface area of a trapezoid with consecutive sides a, c, b, d:

${\displaystyle K={\frac {ane}{ii}}(a+b){\sqrt {c^{2}-{\frac {1}{4}}\left((b-a)+{\frac {c^{two}-d^{2}}{b-a}}\correct)^{two}}}}$

where a and b are parallel and b > a.^[ten] This formula can be factored into a more symmetric version^[6]

${\displaystyle One thousand={\frac {a+b}{4|b-a|}}{\sqrt {(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}.}$

When one of the parallel sides has shrunk to a bespeak (say a = 0), this formula reduces to Heron'south formula for the area of a triangle.

Another equivalent formula for the area, which more closely resembles Heron'south formula, is^[vi]

${\displaystyle Thousand={\frac {a+b}{|b-a|}}{\sqrt {(southward-b)(southward-a)(s-b-c)(south-b-d)}},}$

where ${\displaystyle south={\tfrac {1}{2}}(a+b+c+d)}$ is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula, simply information technology differs from it, in that a trapezoid might not be circadian (inscribed in a circle). The formula is also a special example of Bretschneider's formula for a full general quadrilateral).

From Bretschneider'south formula, it follows that

${\displaystyle 1000={\sqrt {{\frac {(ab^{2}-a^{2}b-advertising^{ii}+bc^{2})(ab^{2}-a^{two}b-ac^{2}+bd^{2})}{(2(b-a))^{2}}}-\left({\frac {b^{2}+d^{2}-a^{two}-c^{2}}{4}}\right)^{two}}}.}$

The line that joins the midpoints of the parallel sides, bisects the area.

Diagonals

The lengths of the diagonals are^[6]

${\displaystyle p={\sqrt {\frac {ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}}},}$

${\displaystyle q={\sqrt {\frac {ab^{2}-a^{2}b-advertizement^{2}+bc^{2}}{b-a}}}}$

where a and b are the bases, c and d are the other ii sides, and a < b.

If the trapezoid is divided into four triangles by its diagonals AC and BD (as shown on the right), intersecting at O, and then the surface area of ${\displaystyle \triangle }$ AOD is equal to that of ${\displaystyle \triangle }$ BOC , and the production 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.^[6]

Permit the trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and DC. Permit E be the intersection of the diagonals, and let F exist on side DA and G be on side BC such that FEG is parallel to AB and CD. And then FG is the harmonic mean of AB and DC:^[11]

${\displaystyle {\frac {1}{FG}}={\frac {1}{2}}\left({\frac {1}{AB}}+{\frac {1}{DC}}\right).}$

The line that goes through both the intersection signal of the extended nonparallel sides and the intersection betoken of the diagonals, bisects each base.^[12]

Other properties

The center of area (centre of mass for a uniform lamina) lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side b given by^[13]

${\displaystyle ten={\frac {h}{3}}\left({\frac {2a+b}{a+b}}\right).}$

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

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

If the angle bisectors to angles A and B intersect at P, and the bending bisectors to angles C and D intersect at Q, then^[12]

${\displaystyle PQ={\frac {|Ad+BC-AB-CD|}{2}}.}$

More than on terminology

The term trapezoid was one time defined as a quadrilateral without whatever parallel sides in Britain and elsewhere. (The Oxford English language Dictionary says "Frequently called by English writers in the 19th century".)^[xv] According to the Oxford English Dictionary, the sense of a effigy with no sides parallel is the meaning for which Proclus introduced the term "trapezoid". This is retained in the French trapézoïde (^[16]), German Trapezoid, and in other languages. However, this particular sense is considered obsolete.

A trapezium in Proclus' sense is a quadrilateral having one pair of its contrary sides parallel. This was the specific sense in England in 17th and 18th centuries, and again the prevalent one in recent use outside North America. A trapezium every bit whatever quadrilateral more general than a parallelogram is the sense of the term in Euclid.

Confusingly, the give-and-take trapezium was sometimes used in England from c. 1800 to c. 1875, to denote an irregular quadrilateral having no sides parallel. This is now obsolete in England, only continues in North America. Notwithstanding this shape is more usually (and less confusingly) just chosen an irregular quadrilateral.^{[17] [xviii]}

Applications

Architecture

In compages the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the pinnacle, in Egyptian style. If these have directly sides and abrupt angular corners, their shapes are normally isosceles trapezoids. This was the standard way for the doors and windows of the Inca.^[xix]

Geometry

The crossed ladders problem is the problem of finding the altitude between the parallel sides of a correct trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection.

Biology

In morphology, taxonomy and other descriptive disciplines in which a term for such shapes is necessary, terms such as trapezoidal or trapeziform commonly are useful in descriptions of particular organs or forms.^[20]

See too

• Polite number, also known equally a trapezoidal number
• Trapezoidal rule, too known as trapezium dominion
• Wedge, a polyhedron defined past two triangles and iii trapezoid faces.

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Source: https://b2b.partcommunity.com/community/knowledge/en/detail/4287/Trapezoid

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