A trapezoid

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. In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. Some authors ^[1] 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 Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides ^[2]

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^[3], but this usage is now obsolete. ^[4] A trapezoid with vertices ABCD would be denoted as 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. An isosceles Trapezoid ( isosceles trapezium in British English) is a Quadrilateral with a line of Symmetry bisecting one pair

If sides AD and BC are also parallel, then the trapezoid is also a parallelogram. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides Otherwise, the other two opposite sides may be extended until they meet at a point, forming a triangle containing the trapezoid. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line

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). ↔ In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called A pair of Angles is supplementary if their measurements add up to 180 degrees If the two supplementary angles are adjacent (i This article describes the unit of angle For other meanings see Degree. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 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. A diagonal can refer to a line joining two nonconsecutive vertices of a Polygon or Polyhedron, or in contexts any upward or downward sloping line A ratio is an expression which compares quantities relative to each other

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. In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided 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. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent 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:

The quantity 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. In Mathematics, an average, or central tendency of a Data set refers to a measure of the "middle" or " expected " value of

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:

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. In Geometry, Heron's (or Hero's formula states that the Area (A of a Triangle whose sides have lengths a, b, and

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. MathWorld is an online Mathematics reference work created and largely written by Eric W at: Animated course (construction, circumference, area)