quadrangle

A quadrangle, in geometry often called a quadrilateral, is a polygon with four sides and four vertices. The term quadrangle is derived from Latin roots meaning four angles, and in everyday language it is sometimes used for a four-sided courtyard or open space enclosed by buildings, especially in university campuses or historic sites.

In pure geometry, a quadrangle can be simple (its edges do not cross) and is typically classified as convex or concave. A convex quadrangle has all interior angles less than 180 degrees, while a concave one has a reflex angle greater than 180 degrees. The sum of the interior angles in any simple quadrilateral is always 360 degrees. The diagonals connect opposite vertices; in a convex quadrangle they intersect inside the figure, whereas in a concave one their intersection may lie outside the polygon.

Area can be determined by dividing the quadrangle into two triangles along a diagonal or by applying the shoelace formula to the vertex coordinates. For cyclic quadrilaterals (those inscribed in a circle), Brahmagupta’s formula gives the area as sqrt((s−a)(s−b)(s−c)(s−d)) where s is the semiperimeter and a, b, c, d are the side lengths.

Quadrangles include many special types, such as parallelograms, rectangles, squares, rhombi, and kites. If a quadrangle has at least one pair of parallel opposite sides, it is often called a trapezoid (US) or trapezium (UK). In geometry and computer graphics, quadrangles are common primitives for modeling and rendering planar surfaces.