Trapezoid (geometry)

A trapezoidal ( latin trapezium of ancient Greek τραπέζιον trapézion , diminutive of τράπεζα trapeza "table", "four legs") is in the geometry of a planar quadrilateral with two parallel mutually facing sides.

General

Trapezoid with corners A, B, C, D,
sides a, b, c, d
and angles α, β, γ, δ.
The following are shown with broken lines:
the height h and
the diagonals AC and BD of the trapezoid and their intersection point S.

The two parallel sides are called the base of the trapezoid. One of these base sides (usually the longer one) is often referred to as the base of the trapezoid, the two adjoining, generally non-parallel sides, often called the legs. In the trapezoid there are two pairs of adjacent supplementary angles , that is, the angles add up to 180  degrees .

The height of the trapezoid is the distance between the two parallel sides. ${\displaystyle h}$

Each convex trapezoid has two diagonals that intersect in equal proportions . The diagonals divide the trapezoid into four triangles , two of which are similar to each other and two of which are equal in area. This can be proven like this:

Let be a convex trapezoid and the intersection of its diagonals (see illustration), then the triangles and are similar to each other because they have the same angles , because these angles are vertex angles and alternating angles in the case of parallels. From the similarity of these two triangles it follows directly that the diagonals intersect in the same ratio , that is . The triangles and are equal in area, because the triangles and are equal in area, because both have the same base and the same height . Only the common triangle needs to be subtracted from both triangles . ${\displaystyle ABCD}$ ${\displaystyle S}$ ${\displaystyle DCS}$ ${\displaystyle ABS}$ ${\displaystyle {\tfrac {DS}{BS}}={\tfrac {CS}{AS}}}$ ${\displaystyle ADS}$ ${\displaystyle BCS}$ ${\displaystyle ABC}$ ${\displaystyle ABD}$ ${\displaystyle ABS}$

A trapezoid is either a convex or an overturned square . Overturned trapezoids are usually not counted among the trapezoids.

Formulas

Mathematical formulas for the trapezoid
Area	${\displaystyle A={\frac {(a+c)\cdot h}{2}}}$
height	${\displaystyle h={\frac {2}{c-a}}\cdot {\sqrt {s\cdot (s+a-c)\cdot (s-b)\cdot (s-d)}}}$(for ), ${\displaystyle a<c}$ With ${\displaystyle s={\frac {b+c+d-a}{2}}}$
	${\displaystyle h={\frac {2}{a-c}}\cdot {\sqrt {s\cdot (s+c-a)\cdot (s-d)\cdot (s-b)}}}$(for ), ${\displaystyle c<a}$ With ${\displaystyle s={\frac {a+b+d-c}{2}}}$
	${\displaystyle h={\frac {2\cdot A}{a+c}}}$
	${\displaystyle h=b\cdot \sin(\beta )=b\cdot \sin(\gamma )=d\cdot \sin(\delta )=d\cdot \sin(\alpha )}$
Length of the diagonal	${\displaystyle e={\sqrt {\frac {c\cdot b^{2}+c^{2}\cdot a-c\cdot a^{2}-a\cdot d^{2}}{c-a}}}}$(for ) ${\displaystyle a<c}$ ${\displaystyle e={\sqrt {\frac {a\cdot d^{2}+a^{2}\cdot c-a\cdot c^{2}-c\cdot b^{2}}{a-c}}}}$(for ) ${\displaystyle c<a}$
	${\displaystyle f={\sqrt {\frac {c\cdot d^{2}+c^{2}\cdot a-c\cdot a^{2}-a\cdot b^{2}}{c-a}}}}$(for ) ${\displaystyle a<c}$ ${\displaystyle f={\sqrt {\frac {a\cdot b^{2}+a^{2}\cdot c-a\cdot c^{2}-c\cdot d^{2}}{a-c}}}}$(for ) ${\displaystyle c<a}$
Interior angle	${\displaystyle \alpha +\delta =\beta +\gamma =180^{\circ }}$

The formula for calculating the height from the side lengths can be derived from the Heronian formula for the triangular area . The relationships for the diagonal lengths are based on the cosine law .

Special cases

Isosceles and symmetrical trapezoid

Isosceles trapezoid with perimeter

Textbooks contain several variants for characterizing an isosceles trapezoid, in particular:

• A trapezoid is called isosceles if the two sides that are not base sides are of the same length.
• A trapezoid is called isosceles if the two interior angles on one of the parallel sides are equal.
• A trapezoid is called isosceles if it has an axis of symmetry perpendicular to one side.

The first characterization also formally includes parallelograms , which are sometimes - if not explicitly - excluded. The last two characterizations are equivalent and in this case the isosceles trapezoid is also called a symmetrical trapezoid because of the axis symmetry . The interior angles are therefore the same on both parallel sides. The two diagonals are the same length in the symmetrical trapezoid.

The corner points of a symmetrical trapezoid lie on a circle , the circumference of the trapezoid. The trapezoid is thus a quadrilateral chord of this circle. The circumcenter is the intersection of the perpendiculars of the sides of the trapezoid. The trapezoid is divided into two mirror-symmetrical parts from the height that goes through the center of the circumference . ${\displaystyle k}$ ${\displaystyle h}$ ${\displaystyle M}$

A trapezoid that has two of the properties right-angled, point-symmetrical (parallelogram) and axis-symmetrical also automatically has the third and is therefore a rectangle.

Right-angled trapezoid

Right-angled trapezoid

A trapezoid is called right-angled (or orthogonal ) if it has at least one right interior angle . Since all angles in a trapezoid lie on one of the parallel base sides, a right-angled trapezoid must always have at least two right angles that are next to each other. A rectangle is the special case of a right-angled trapezoid. It even has four right interior angles.

Overturned or crossed trapezoid

Overturned trapezoid

In the case of an overturned or crossed trapezoid , it is not the ends of the base sides on the same side that are connected by the other sides, but the opposite sides. So these sides cross in the center of the trapezoid. An overturned trapezoid can be imagined as the square that is formed from the base and the diagonals of a convex trapezoid. The two faces are triangles that are similar to one another . Overturned trapezoids are usually not counted among the (normal or "real") trapezoids. ${\displaystyle M}$

The area of ​​the crossed trapezoid, i.e. the sum of the areas of the two triangles , is calculated as follows:

${\displaystyle A={\frac {h}{2}}\cdot {\frac {a^{2}+c^{2}}{a+c}}}$

Crossed, right-angled trapezoid

Overturned or crossed trapezoids, which are also at right angles, are used in geodesy to calculate surface areas , for example from orthogonal recordings . They consist of two right triangles that touch at one corner . The difference between the areas of the two triangles results in

${\displaystyle A_{\Delta }=A_{D_{1}}-A_{D_{2}}=h{\frac {a-c}{2}}}$

with . This area is signed. This means that case distinctions are no longer required when calculating areas using the Gaussian trapezoidal formula if a peripheral side of the area intersects the reference line. ${\displaystyle h={\overline {BC}}}$

Concept history

The restriction of the term to quadrilaterals with two parallel sides is relatively recent. Until the beginning of the 20th century, a trapezoid was usually a square in which no pair of sides was parallel, i.e. an irregular square without special properties. For the trapezoid with two parallel sides, the designation parallel trapezoid was common. This use was derived from Euclid's classification of the quadrilaterals, whereby the latter did not consider a quadrilateral with a parallel pair of sides separately, but counted it to the quadrangles without special properties. That is, the trapezoid in Euclid included both the trapezoid and the parallel trapezoid in the above sense. Euclid's exact classification was as follows:

“Among the four-sided figures, that one is called a square ( τετράγωνον ) which is equilateral and right-angled; a rectangle ( ὀρθογώνιον ), which is right-angled but not equilateral; a rhombus (ῥόμβος), which is equilateral but not rectangular; and a rhomboid ( ῥομβοειδὲς σχῆμα ), whose opposite sides and angles are the same, but which is neither equilateral nor right-angled. Every other four-sided figure is called a trapezoid ( τραπέζιον ). "

- Euclid : Elements Book I, 22

In contrast, Proklos , Heron and Poseidonios used the term trapezoid in the modern sense, i.e. for the parallel trapezoid . They called the irregular square a trapezoid ( τραπεζοειδῆ ). This distinction (Engl. Trapezoidal trapezium ) and trapezoid are there so in German and British English. In American English, the terms trapezium and trapezoid are confusingly used in reverse.

Most medieval mathematicians from Boethius onwards adopted Euclid's use of the term as an irregular square . The distinction according to Poseidonios was only rarely taken up again. They have only been found more frequently since the 18th century, e.g. B. Legendre and Thibaut . Jean Henri van Swinden used the term "trapezoid" in Euclid's sense and called the square with two parallel sides parallel trapezoid .

Web links

Commons : Trapeze  - collection of images, videos and audio files

• Eric W. Weisstein : Trapezoid . In: MathWorld (English).
• Eric W. Weisstein : Trapezium . In: MathWorld (English).

Individual evidence

1. ↑ In τράπεζα is a short form of τετράπεζα tetrapeza "four legs" ( τέτρα tetra "four"; πέζα peza "foot"). Compare Karl Menninger : number and number. A cultural history of the number. Vandenhoeck & Ruprecht, 1979, ISBN 3-525-40725-4 . P. 190 ( excerpt (Google) )
2. ↑ ^{a b} Ilja N. Bronstein, Konstantin A. Semendjajew: Taschenbuch der Mathematik . 24th edition. Harri Deutsch, Thun and Frankfurt am Main 1989, ISBN 3-87144-492-8 , pp. 192 .
3. ↑ ^{a b} Federal Mathematics Competition: Exercises and Solutions, 1st Round 2012 . S. 8 ( PDF ).
4. ^ Friedrich Zech: Basic course in mathematics didactics . 10th edition. Beltz, Weinheim and Basel 2002, ISBN 3-407-25216-1 , p. 256 .
5. ^ Student dudes: Mathematik I. Dudenverlag, 8th edition, Mannheim 2008, p. 457.
6. ↑ In Bronstein / Semendjajew the isosceles trapezoid is characterized by the length of the legs, but the formula given below does not apply to parallelograms. In the solutions of the Federal Mathematics Competition 2012, the characterizations using side lengths and interior angles are named as alternatives. They are only equivalent if parallelograms are excluded in the first case.
7. ^ Pierer's Universal Lexicon. 4th edition 1857–1865, article “Trapez” .
8. ^ Meyer's Large Conversational Lexicon. 6th edition 1905–1909, article “Paralleltrapēz” .
9. ↑ It was a “real” parallelogram: a parallelogram that is neither a rhombus nor a rectangle (and therefore certainly not a square).
10. ↑ Euclid's elements. Original Greek text.
11. ↑ English translation of Euclid's Elements (Book I, Definition 22) with annotations.
12. ↑ ^{a b} Johannes Tropfke: History of Elementary Mathematics. Volume 4: Plane Geometry. de Gruyter, 1940 ( f. # v = onepage restricted preview in the Google book search).