Isosceles trapezoid

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In
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method consists in assuming a smal ...
, an isosceles trapezoid (isosceles trapezium in
British English British English (BrE) is the standard dialect of the English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the ...
) is a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope, ...
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and t ...

with a line of
symmetry Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

bisecting one pair of opposite sides. It is a special case of a
trapezoid In Euclidean geometry, a Convex polygon, convex quadrilateral with at least one pair of parallel (geometry) , parallel sides is referred to as a trapezium () in English outside North America, but as a trapezoid () in American English, America ...

. Alternatively, it can be defined as a
trapezoid In Euclidean geometry, a Convex polygon, convex quadrilateral with at least one pair of parallel (geometry) , parallel sides is referred to as a trapezium () in English outside North America, but as a trapezoid () in American English, America ...

in which both legs and both base angles are of equal measure. Note that a non-rectangular
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides (the bases) are
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
, and the two other sides (the legs) are of equal length (properties shared with the
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method con ...

). The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the
supplementary angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakod ...
of a base angle at the other base).

# Special cases

Rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

s and
square In Euclidean geometry, a square is a regular polygon, regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree (angle), degree angles, π/2 radian angles, or right angles). It can also be defined as a rec ...

s are usually considered to be special cases of isosceles trapezoids though some sources would exclude them. Another special case is a ''3-equal side trapezoid'', sometimes known as a ''trilateral trapezoid'' or a ''trisosceles trapezoid''. They can also be seen dissected from
regular polygon In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method cons ...
s of 5 sides or more as a truncation of 4 sequential vertices.

## Self-intersections

Any non-self-crossing
quadrilateral A quadrilateral is a polygon in Euclidean geometry, Euclidean plane geometry with four Edge (geometry), edges (sides) and four Vertex (geometry), vertices (corners). Other names for quadrilateral include quadrangle (in analogy to triangle) and t ...

with exactly one axis of symmetry must be either an isosceles trapezoid or a
kite . This sparless, ram-air inflated kite, has a complex bridle formed of many strings attached to the face of the wing. A kite is a tethered heavier-than-air or lighter-than-air craft with wing A wing is a type of fin that produces lift wh ...
.. However, if crossings are allowed, the set of symmetric quadrilaterals must be expanded to include also the crossed isosceles trapezoids, crossed quadrilaterals in which the crossed sides are of equal length and the other sides are parallel, and the
antiparallelogram In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

s, crossed quadrilaterals in which opposite sides have equal length. Every
antiparallelogram In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

has an isosceles trapezoid as its
convex hull In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

, and may be formed from the diagonals and non-parallel sides of an isosceles trapezoid.

# Characterizations

If a quadrilateral is known to be a
trapezoid In Euclidean geometry, a Convex polygon, convex quadrilateral with at least one pair of parallel (geometry) , parallel sides is referred to as a trapezium () in English outside North America, but as a trapezoid () in American English, America ...

, it is ''not'' sufficient just to check that the legs have the same length in order to know that it is an isosceles trapezoid, since a
rhombus In plane Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's m ...

is a special case of a trapezoid with legs of equal length, but is not an isosceles trapezoid as it lacks a line of symmetry through the midpoints of opposite sides. Any one of the following properties distinguishes an isosceles trapezoid from other trapezoids: *The diagonals have the same length. *The base angles have the same measure. *The segment that joins the midpoints of the parallel sides is perpendicular to them. *Opposite angles are supplementary, which in turn implies that isosceles trapezoids are
cyclic quadrilateral In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rako ...

s. *The diagonals divide each other into segments with lengths that are pairwise equal; in terms of the picture below, , (and if one wishes to exclude rectangles).

# Angles

In an isosceles trapezoid, the base angles have the same measure pairwise. In the picture below, angles ∠''ABC'' and ∠''DCB'' are obtuse angles of the same measure, while angles ∠''BAD'' and ∠''CDA'' are
acute angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakod ...
s, also of the same measure. Since the lines ''AD'' and ''BC'' are parallel, angles adjacent to opposite bases are , that is, angles

# Diagonals and height

350px, Another isosceles trapezoid. The
diagonal Image:Cube diagonals.svg, The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length \sqrt 3, while AC (shown in red) is a face diagonal and has length \sqrt 2. In geometry, a diagonal is a line segment joinin ...
s of an isosceles trapezoid have the same length; that is, every isosceles trapezoid is an
equidiagonal quadrilateral In Euclidean geometry, an equidiagonal quadrilateral is a convex polygon, convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified firs ...

. Moreover, the diagonals divide each other in the same proportions. As pictured, the diagonals ''AC'' and ''BD'' have the same length () and divide each other into segments of the same length ( and ). The
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ...

in which each diagonal is divided is equal to the ratio of the lengths of the parallel sides that they intersect, that is, :$\frac = \frac = \frac.$ The length of each diagonal is, according to
Ptolemy's theorem In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose Vertex (geometry)#Of a polytope, vertices lie on a common circle). The theorem is named after the Roma ...

, given by :$p=\sqrt$ where ''a'' and ''b'' are the lengths of the parallel sides ''AD'' and ''BC'', and ''c'' is the length of each leg ''AB'' and ''CD''. The height is, according to the
Pythagorean theorem In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

, given by :$h=\sqrt=\tfrac\sqrt.$ The distance from point ''E'' to base ''AD'' is given by :$d=\frac$ where ''a'' and ''b'' are the lengths of the parallel sides ''AD'' and ''BC'', and ''h'' is the height of the trapezoid.

# Area

The area of an isosceles (or any) trapezoid is equal to the average of the lengths of the base and top (''the parallel sides'') times the height. In the adjacent diagram, if we write , and , and the height ''h'' is the length of a line segment between ''AD'' and ''BC'' that is perpendicular to them, then the area ''K'' is given as follows: :$K=\frac\left\left(a+b\right\right).$ If instead of the height of the trapezoid, the common length of the legs ''AB'' =''CD'' = ''c'' is known, then the area can be computed using
Brahmagupta's formula In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakodī; ...
for the area of a cyclic quadrilateral, which with two sides equal simplifies to :$K = \sqrt,$ -where $s = \tfrac\left(a + b + 2c\right)$ is the semi-perimeter of the trapezoid. This formula is analogous to
Heron's formula In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

to compute the area of a triangle. The previous formula for area can also be written as :$K= \frac \sqrt.$

The radius in the circumscribed circle is given byTrapezoid at Math24.net: Formulas and Table

Accessed 1 July 2014.
:$R=c\sqrt.$ In a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a para ...

where ''a'' = ''b'' this is simplified to $R=\tfrac\sqrt$.