Trapezoid

In geometry, a trapezoid is (from the ancient Greek. Τράπεζα trapeza "table" ) a plane rectangle lying with two parallel sides.

General

The two parallel sides are called bases of the trapezoid. One of these base sides (usually longer ) is often referred to as the base of the trapezium, the two adjacent ( generally non-parallel ) sides often a leg. The trapezoid, there are two pairs of adjacent supplementary angle ( ie the angle complementary to 180 degrees).

The height of the trapezoid is the distance between the two parallel sides.

Each (convex ) trapezoid has two diagonals which intersect each other in the same ratio. The diagonals divide the trapezoid into four triangles, two of which are similar to each other and two coextensive. Proof: Let ABCD be a convex trapezoidal and S is the intersection of its diagonals ( see drawing), then the triangles DCS and ABS are similar because they have the same angle ( apex angles and alternate angles at parallels ). From the similarity of these two triangles then follows directly that the diagonals intersect each other in the same ratio, that is. The triangles ADS and BCS are coextensive, because the triangles ABC and ABD are equal in area (both have the same base line and same height). From the two triangles the common triangle ABS only still needs to be deducted.

A trapezoid is either a convex or has rolled square. About Beaten trapezoids are not normally counted among the ( normal or "real" ) trapezoids.

Formulas

( a < c ),

With

( for c < a),

With

The formula for calculating the height of the side lengths can be derived from the African hero formula for the triangular surface. The relations for the diagonal lengths are based on the law of cosines.

Special cases

Isosceles and symmetrical trapezium

In textbooks, several variants can be used to characterize an isosceles trapezoid, in particular:

The first characterization includes formal and parallelograms with one, but sometimes - be excluded - even if not explicitly. The last two are equivalent characterizations and in this case the isosceles trapezoid is also called symmetrical trapezium because the axis of symmetry. Therefore, the interior angles are on the two parallel sides of equal size. The two diagonals are of equal length in the symmetric trapezoid.

The vertices of a symmetric trapezoid lie on a circle k, the perimeter of the trapezoid. The trapeze is therefore a cyclic quadrilateral this circle. The circumcenter is the intersection of the perpendicular bisectors of the sides of the trapezoid. The trapeze is divided by the height h, which passes through the circumcenter M, in two mirror- symmetric parts.

Right-angled trapezium

A trapezoid is perpendicular (or even orthogonal) if it has at least one right-hand internal angle. As applied in a trapezoidal all angles to one of the parallel bases of a right-angled trapezium must always have at least two right angles set next to each other. A rectangle is a special case of a right-angled trapezium; it even has four right interior angle.

About Proposed or entangled trapezoidal

When you strike or crossed over the equilateral trapezoidal ends of the base sides are not connected through the other pages, but the opposite. These sites therefore intersect at the center M of the trapezoid. One can get an over proposed Keystone thought of as the square, which is formed from the base sides and the diagonals of a convex trapezoid. The two faces are similar to each other triangles. About Beaten trapezoids are not normally counted among the ( normal or "real" ) trapezoids.

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

About Chipped or folded trapezoids addition at right angles, are used in geodesy to calculate surface areas, such as Orthogonalaufnahmen. They consist of two right-angled triangles which touch at one corner. The difference between the areas of the two triangles is given by

With. This area is signed. This eliminates the need for area calculations using the Gaussian trapezoidal rule case distinction when a Umringsseite the surface intersects the reference line.

Conceptual history

The restriction of the term to quadrilaterals with two parallel sides is relatively young. Until the beginning of the 20th century was called the keystone usually a quadrilateral in which no pair of sides is parallel. So already used the concept of Euclid. Of the trapezoid with two parallel sides, the term trapezoid was common. Euclid used the following classification of quadrilaterals:

" In the four-sided figures is that of a square ( τετράγωνον ) which is equal to each other and at right angles to, a rectangle ( ὀρθογώνιον ) which, although a right angle, but not equilateral, a rhombus ( ῥόμβος ), which is indeed equal to one another, but not perpendicular, a rhomboid ( ῥομβοειδὲς ) whose mutually opposite sides and angles are the same, but which is neither equilateral or right-angled. Any other four -sided figure hot trapezoid ( τραπέζιον ). "

In contrast used Proclus, Heron and Posidonius the term trapezoid in the modern sense of the trapezoid; the irregular quadrilateral they were trapezoid ( τραπεζοειδῆ ). This distinction of trapezoidal (English trapezium ) and trapezoid there are so in German and British English. In American English, the terms trapezium and trapezoid confusingly vice versa.

Most mathematicians of the Middle Ages from Boethius took Euclid's use of the term as a general quadrangle. The distinction according to Posidonius was rarely taken up again. Only since the 18th century they are found frequently, eg Legendre and Thibaut. Jean Henri van Swinden used the term trapezoid in the sense of Euclid and named the quadrilateral with two parallel sides Trapezoid.