Dictionary Definition
trapezoid
Noun
1 a quadrilateral with two parallel sides
2 the wrist bone between the trapezium and the
capitate bones [syn: trapezoid
bone, os
trapezoideum]
User Contributed Dictionary
English
Pronunciation
- , /ˈtræpəzɔɪd/, /"tr
Extensive Definition
A trapezoid (in North America)
or a trapezium (in Britain and elsewhere) is a quadrilateral, which is
defined as a closed shape with four linear sides, that has one pair
of parallel
lines for sides. Some authors define it as a quadrilateral having
exactly one pair of parallel sides, so as to exclude parallelograms, which
otherwise would be regarded as a special type of trapezoid, but
most mathematicians use the inclusive definition.
In North America, the term
trapezium is used to refer to a quadrilateral with no parallel
sides. The term trapezoid was once defined as a quadrilateral
without any parallel sides in Britain and elsewhere, but this usage
is now obsolete. A trapezoid with vertices ABCD would be denoted as
trapezoidnotation ABCD.
Characteristics and properties
In an isosceles trapezoid, the base angles are equal, and so are the other pair of opposite sides AD and BC.If sides AD and BC are also
parallel, then the trapezoid is also a parallelogram. Otherwise,
the other two opposite sides may be extended until they meet at a
point, forming a triangle
containing the trapezoid.
A quadrilateral is a trapezoid
if
and only if it contains two adjacent angles that are supplementary,
that is, they add up to one straight angle of 180 degrees
(π radians). Another necessary and
sufficient condition is that the diagonals cut each other in
mutually the same ratio;
this ratio is the same as that between the lengths of the parallel
sides.
The mid-segment (occasionally
referred to as the median) of a trapezoid is the segment that joins
the midpoints of the other pair of opposite sides. It is parallel
to the two parallel sides, and its length is the arithmetic
mean of the lengths of those sides. The line joining the
mid-points of the parallel sides (which could also be called the
median) bisects the area.
The area of a trapezoid can be computed
as the length of the mid-segment, multiplied by the distance along
a perpendicular
line between the parallel sides. This yields as a special case the
well-known formula for the area of a triangle, by considering a
triangle as a degenerate trapezoid in which one of the parallel
sides has shrunk to a point.
Thus, if a and b are the two
parallel sides and h is the distance (height) between the
parallels, the area formula is as follows:
- A= h\frac.
The quantity \frac is the
average of the
horizontal lengths of the trapezoid, so the area can be understood
to be the product of the height and average length of the
shape.
Another formula for the area
can be used when all that is known are the lengths of the four
sides. If the sides are a, b, c and d, and a and c are parallel
(where a is the longer parallel side), then:
- A=\frac\sqrt.
This formula does not work
when the parallel sides a and c are equal since we would have
division by zero. In this case the trapezoid is necessarily a
parallelogram (and so b = d) and the numerator of the formula would
also equal zero. In fact, the sides of a parallelogram aren't
enough to determine its shape or area, the area of a parallelogram
with sides a and b can be any number from "a b" to
"zero".
When the smaller parallel side
c is set to zero, this formula turns to be Heron's
formula.
If the trapezoid above is
divided into 4 triangles by its diagonals AC and BD, intersecting
at O, then the area of ΔAOD is equal to that of
ΔBOC, and the product of the areas of ΔAOD and
ΔBOC is equal to that of ΔAOB and
ΔCOD. The ratio of the areas of each pair of adjacent
triangles is the same as that between the lengths of the parallel
sides.
References
External links
- "Trapezoid" on MathWorld
- Trapezoid definition Area of a trapezoid Median of a trapezoid With interactive animations
- Trapezoid (North America) at elsy.at: Animated course (construction, circumference, area)
trapezoid in Asturian:
Trapeciu (figura)
trapezoid in Azerbaijani:
Trapesiya
trapezoid in Bulgarian:
Трапец
trapezoid in Catalan:
Trapezi
trapezoid in Czech:
Lichoběžník
trapezoid in Danish: Trapez
(matematik)
trapezoid in German: Trapez
(Geometrie)
trapezoid in Estonian:
Trapets
trapezoid in Modern Greek
(1453-): Τραπέζιο
trapezoid in Spanish:
Trapecio (geometría)
trapezoid in Esperanto:
Trapezo
trapezoid in French:
Trapèze
trapezoid in Korean:
사다리꼴
trapezoid in Croatian: Trapez
(geometrija)
trapezoid in Italian:
Trapezio (geometria)
trapezoid in Hebrew:
טרפז
trapezoid in Georgian:
ტრაპეცია
trapezoid in Latvian:
Trapece
trapezoid in Lithuanian:
Trapecija
trapezoid in Hungarian:
Trapéz
trapezoid in Dutch:
Trapezium
trapezoid in Japanese:
台形
trapezoid in Polish:
Trapez
trapezoid in Portuguese:
Trapézio (geometria)
trapezoid in Romanian:
Trapez
trapezoid in Quechua: Pilla
t'asra
trapezoid in Russian:
Трапеция
trapezoid in Slovenian:
Trapez
trapezoid in Serbian: Трапез
(геометрија)
trapezoid in Finnish:
Puolisuunnikas
trapezoid in Swedish:
Parallelltrapets
trapezoid in Tamil:
சரிவகம்
trapezoid in Vietnamese: Hình
thang
trapezoid in Ukrainian:
Трапеція
trapezoid in Vlaams:
Trapezium
trapezoid in Yiddish:
טראפעז
trapezoid in Chinese:
梯形
Synonyms, Antonyms and Related Words
cube-shaped, cubed, cubic, cubiform, cuboid, diced, foursquare, oblong, orthogonal, quadrangular, quadrate, quadriform, quadrilateral, rectangular, rhombic, rhomboid, square, tetragonal, tetrahedral, trapezohedral