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Trapezoid (USA definition)
From Greek: trapeze, oid "like"
A quadrilateral which has at least one pair of parallel sides
(but see definition notes below)
Try this Drag the orange dots on each vertex
to reshape the figure. Notice that the sides AB and CD remain parallel.
Attributes
Base 
One of the parallel sides. Every trapezoid has two bases. See
Base definitions.

Leg 
The sides AC and BD above are called the legs of the trapezoid, and are usually not parallel,
although they could be (see parallelogram note below). Every trapezoid has two legs.

Altitude 
The altitude of a trapezoid is the
perpendicular distance from one base to the other.
(One base may need to be extended).

Median 
The median
of a trapezoid is a line joining the midpoints of the two legs. See Trapezoid median

Area 
The usual way to calculate the area is the average base length times altitude.
See Area of a Trapezoid

Perimeter 
The distance a round the trapezoid. The sum of its side lengths.
See Perimeter of a Trapezoid

If both legs are the same length, this is called an isosceles trapezoid, and both base angles are the same.
If the legs are parallel, it now has two pairs of parallel sides, and is a parallelogram.
Coordinate Geometry
In coordinate geometry, if you know the coordinates of the four vertices,
you can calculate various properties of it, including the altitude and median.
For more on this, see Trapezoid definition (Coordinate Geometry)
Definition notes
There is considerable confusion over the definition of 'trapezoid' and 'trapezium' due to differences in the British and US versions.
As you can see from the table below, the meanings of the two words are exactly reversed between the US and British interpretations.

British 
USA 
Trapezoid 
A quadrilateral with no sides parallel 
A quadrilateral with one pair of parallel sides 
Trapezium 
A quadrilateral with one pair of parallel sides 
A quadrilateral with no sides parallel 
For the definition of trapezium see Trapezium
Related polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
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