Trapezoid


In geometry, a trapezoid in North American English, or trapezium in British English, is a quadrilateral that has at least one pair of parallel sides.
The parallel sides are called the bases of the trapezoid. The other two sides are called the legs or lateral sides. If the trapezoid is a parallelogram, then the choice of bases and legs is arbitrary.
A trapezoid is usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases. If shape ABCD is a convex trapezoid, then ABDC is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids.

Definitions

Trapezoid can be defined exclusively or inclusively. Under an exclusive definition a trapezoid is a quadrilateral having pair of parallel sides, with the other pair of opposite sides non-parallel. Parallelograms including rhombi, rectangles, and squares are then not considered to be trapezoids. Under an inclusive definition, a trapezoid is any quadrilateral with one pair of parallel sides. In an inclusive classification scheme, definitions are hierarchical: a square is a type of rectangle and a type of rhombus, a rectangle or rhombus is a type of parallelogram, and every parallelogram is a type of trapezoid. A trapezoid can also be defined as any quadrilateral that is not a parallelogram; this definition is also exclusive and is used in Euclid's Elements.
Professional mathematicians and post-secondary geometry textbooks nearly always prefer inclusive definitions and classifications, because they simplify statements and proofs of geometric theorems. In primary and secondary education, definitions of rectangle and parallelogram are also nearly always inclusive, but an exclusive definition of trapezoid is commonly found. This article uses the inclusive definition and considers parallelograms to be special kinds of trapezoids.
To avoid confusion, some sources use the term proper trapezoid to describe trapezoids with exactly one pair of parallel sides, analogous to uses of the word proper in some other mathematical objects.

Etymology

In the ancient Greek geometry of Euclid's Elements, quadrilaterals were classified into exclusive categories: square; oblong ; rhombus; rhomboid, meaning a non-rhombus non-rectangle parallelogram; or trapezium, meaning any quadrilateral not already included in the previous categories.
The Neoplatonist philosopher Proclus wrote an influential commentary on Euclid with a richer set of categories, which he attributed to Posidonius. In this scheme, a quadrilateral can be a parallelogram or a non-parallelogram. A parallelogram can itself be a square, an oblong, a rhombus, or a rhomboid. A non-parallelogram can be a trapezium with exactly one pair of parallel sides, which can be isosceles or scalene ; or a trapezoid with no parallel sides.
All European languages except American English follow Proclus's meanings of trapezium and trapezoid,. However, the meaning in British English was reversed for much of the 19th century. In 1795, an influential mathematical dictionary published by Charles Hutton transposed the two terms without explanation, leading to widespread inconsistency. Hutton's change was reverted in British English in about 1875, but it has been retained in American English to the present. Late 19th century American geometry textbooks define a trapezium as having no parallel sides, a trapezoid as having exactly one pair of parallel sides, and a parallelogram as having two sets of opposing parallel sides.
To avoid confusion between contradictory British and American meanings of trapezium and trapezoid, quadrilaterals with no parallel sides have sometimes been called irregular quadrilaterals.

Special cases

An isosceles trapezoid is a trapezoid where the base angles have the same measure. As a consequence the two legs are also of equal length and it has reflection symmetry. This is possible for acute trapezoids or right trapezoids as rectangles. An acute trapezoid is a trapezoid with two adjacent acute angles on its longer base, and the isosceles trapezoid is an example of an acute trapezoid. The isosceles trapezoid has a special case known as a three-sided trapezoid, meaning it is a trapezoid wherein two trapezoid's legs have equal lengths as the trapezoid's base at the top. The isosceles trapezoid is the convex hull of an antiparallelogram, a type of crossed quadrilateral. Every antiparallelogram is formed with such a trapezoid by replacing two parallel sides by the two diagonals.
An obtuse trapezoid, on the other hand, has one acute and one obtuse angle on each base. An example is parallelogram with equal acute angles.
A right trapezoid is a trapezoid with two adjacent right angles. One special type of right trapezoid is by forming three right triangles, which was used by James Garfield to prove the Pythagorean theorem.
A tangential trapezoid is a trapezoid that has an incircle.

Condition of existence

Four lengths a, c, b, d can constitute the consecutive sides of a non-parallelogram trapezoid with a and b parallel only when
The quadrilateral is a parallelogram when, but it is an ex-tangential quadrilateral when.

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 up to 180 degrees.
  • The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.
  • The diagonals cut each other in mutually the same ratio.
  • The diagonals cut the quadrilateral into four triangles of which one opposite pair have equal areas.
  • 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.
  • The areas S and T of some two opposite triangles of the four triangles formed by the diagonals satisfy the equation
  • The midpoints of two opposite sides of the trapezoid and the intersection of the diagonals are collinear.
  • The angles in the quadrilateral ABCD satisfy
  • The cosines of two adjacent angles sum to 0, as do the cosines of the other two angles.
  • The cotangents of two adjacent angles sum to 0, as do the cotangents of the other two adjacent angles.
  • One bimedian divides the quadrilateral into two quadrilaterals of equal areas.
  • Twice the length of the bimedian connecting the midpoints of two opposite sides equals the sum of the lengths of the other sides.
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
  • The distance v between the midpoints of the diagonals satisfies the equation

    Properties

Midsegment and height

The midsegment or median of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases. Its length m is equal to the average of the lengths of the bases a and b of the trapezoid,
The midsegment of a trapezoid is one of the two bimedians.
The height is the perpendicular distance between the bases. In the case that the two bases have different lengths, the height of a trapezoid h can be determined by the length of its four sides using the formula
where c and d are the lengths of the legs and.

Area

The area of a trapezoid is given by the product of the midsegment and the height:
where and are the lengths of the bases, and is the height. This method was used in the classical age of India in Aryabhata's Aryabhatiya, yielding 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.
The 7th-century Indian mathematician Bhāskara I derived the following formula for the area of a trapezoid with consecutive sides,,, :
where and are parallel and. This formula can be factored into a more symmetric version
When one of the parallel sides has shrunk to a point, this formula reduces to Heron's formula for the area of a triangle.
Another equivalent formula for the area, which more closely resembles Heron's formula, is
where is the semiperimeter of the trapezoid..
From Bretschneider's formula, it follows that
The bimedian connecting the parallel sides bisects the area. More generally, any line drawn through the midpoint of the median parallel to the bases, that intersects the bases, bisects the area. Any triangle connecting the two ends of one leg to the midpoint of the other leg is also half of the area.

Diagonals

The lengths of the diagonals are
where is the short base, is the long base, and and are the trapezoid legs.
If the trapezoid is divided into four triangles by its diagonals AC and BD, intersecting at O, then the area of is equal to that of, and the product of the areas of and is equal to that of and. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.
If is the length of the line segment parallel to the bases, passing through the intersection of the diagonals, with one endpoint on each leg, then is the harmonic mean of the lengths of the bases:
The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.

Other properties

The center of area lies along the line segment joining the midpoints of the parallel sides, at a perpendicular distance x from the longer side b given by
The center of area divides this segment in the ratio
If the angle bisectors to angles A and B intersect at P, and the angle bisectors to angles C and D intersect at Q, then