Trapezoid
A quadrilateral with at least one pair of parallel sides is called, in American and Canadian English, a trapezoid (). In British and other forms of English, a (North American) trapezoid is called a trapezium (). This article uses North American nomenclature. The transposition of these two terms was a result of an error in Charles Hutton's mathematical dictionary A trapezoid is necessarily a convex quadrilateral in Euclidean geometry. The parallel sides are called the "bases" of the trapezoid. The other two sides are called the "legs" (or the "lateral sides") if they are not parallel; otherwise, the trapezoid is  a parallelogram, and there are two pairs of bases). A "scalene trapezoid" is a trapezoid with no sides of equal measure, in contrast with the special cases below. Etymology and "trapezium" vs "trapezoid". Ancient Greek mathematician Euclid defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and the last did not have two sets of parallel sides – a τραπέζια ("trapezia" literally "a table", itself from τετράς ("tetrás"), "four" + πέζα ("péza"), "a foot; end, border, edge"). Two types of "trapezia" were introduced by Proclus (412 to  485 AD) in his commentary on the first book of Euclid's Elements: All European languages follow Proclus's structure as did English until the late 18th century, until an influential mathematical dictionary published by Charles Hutton in 1795 supported without explanation a transposition of the terms. This mistake was corrected in British English in about 1875, but was retained in American English into the modern day. This article uses the term "trapezoid" in the sense that is current in the United States and Canada. The shape is often called an irregular quadrilateral. Inclusive vs exclusive definition. There is some  disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids. Some define a trapezoid as a quadrilateral having "only" one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Others define a trapezoid as a quadrilateral with "at least" one pair of parallel sides (the inclusive definition), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals. Under  the inclusive definition, all parallelograms (including rhombuses, rectangles and squares) are trapezoids. Rectangles have mirror symmetry on mid-edges; rhombuses have mirror symmetry on vertices, while squares have mirror symmetry on both mid-edges and vertices. Special cases. A right trapezoid (also called "right-angled trapezoid") has two adjacent right angles. Right trapezoids are used in the trapezoidal rule for estimating areas under a curve. An acute trapezoid has two adjacent acute angles on its longer "base" edge, while an obtuse trapezoid has one acute and one obtuse angle on each "base". 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 (rectangles). A parallelogram is a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold rotational symmetry (or point reflection symmetry). It is possible for obtuse trapezoids or right trapezoids (rectangles). A tangential trapezoid is a trapezoid that has an incircle. A Saccheri quadrilateral is similar to a trapezoid in the hyperbolic plane, with two adjacent right angles, while it is a rectangle in the Euclidean  plane. A Lambert quadrilateral in the hyperbolic plane has 3 right angles. 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 formula_2, but it is an ex-tangential quadrilateral (which is not a trapezoid) when formula_3. Characterizations. Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid: Additionally, the following properties are equivalent, and each implies that opposite sides "a" and "b" are parallel: Midsegment and height. The "midsegment  (also called the median or midline) 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 other bimedian divides the trapezoid into equal areas). The "height" (or altitude) is the perpendicular distance between the bases. In the case that the two bases have different lengths ("a" ≠ "b"), 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. Area. The area "K" of a trapezoid is given by where "a" and "b" are the lengths of the parallel sides, "h" is the height (the perpendicular distance between these sides), and "m" is the arithmetic mean of the lengths of the two parallel sides. In 499 AD Aryabhata, a great mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, used this method in the "Aryabhatiya" (section 2.8). 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. The 7th-century Indian mathematician Bhāskara I derived the following formula for the area of a trapezoid with consecutive sides "a", "c", "b", "d": where "a" and "b" are parallel and "b" > "a". This formula can be factored into a more symmetric version When one of the parallel sides has shrunk to a point (say "a" = 0), 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 formula_14 is the semiperimeter of the trapezoid. (This formula is similar to Brahmagupta's formula, but it differs from it, in that a trapezoid might not be cyclic (inscribed in a circle). The formula is also a special case of Bretschneider's formula for a general quadrilateral). From Bretschneider's formula, it follows that The line that joins the midpoints of the parallel sides, bisects the area. Diagonals. The lengths of the diagonals are where "a" is the short base, "b" is the long base, and "c" and  d are the trapezoid legs. If the trapezoid is divided into four triangles by its diagonals "AC" and "BD" (as shown on the right), 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. Let the trapezoid have vertices "A", "B", "C", and "D" in sequence and have parallel sides "AB" and "DC". Let "E" be the intersection of the diagonals , and let "F" be on side "DA" and "G" be on side "BC" such that "FEG" is parallel to "AB" and "CD". Then "FG" is the harmonic mean of "AB" and "DC": 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 (center of mass for a uniform lamina) 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 (when taken from the short to the long side) 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 Applications. Architecture. In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering toward the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually isosceles trapezoids. This was the standard style for the doors and windows of the Inca. Geometry. The crossed ladders problem is the  problem of finding the distance between the parallel sides of a right trapezoid, given the diagonal lengths and the distance from the perpendicular leg to the diagonal intersection. Biology. In morphology, taxonomy and other descriptive disciplines in which a term for such shapes is necessary, terms such as "trapezoidal" or "trapeziform" commonly are useful in descriptions of particular organs or forms. Computer engineering. In computer engineering, specifically digital logic and computer architecture, trapezoids are typically utilized to symbolize multiplexors. Multiplexors are logic elements that select between multiple elements and produce a single output based on a select signal. Typical designs  will employ trapezoids without specifically stating they are multiplexors as they are universally equivalent.