being between sides and , and being between sides and , is In the case of a cyclic quadrilateral, the latter formula becomes In a parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to Alternatively, we can write the area in terms of the sides and the intersection angle of the diagonals, as long is not : In the case of a parallelogram, the latter formula becomes Another area formula including the sides , , , is where is the distance between the midpoints of the diagonals, and is the angle between the bimedians. The last trigonometric area formula including the sides , , , and the angle (between and ) is: which can also be used for the area of a concave quadrilateral (having the concave part opposite to angle ), by just changing the first sign to . Non-trigonometric formulas The following two formulas express the area in terms of the sides , , and , the semiperimeter , and the diagonals , : The first reduces to Brahmagupta's formula in the cyclic quadrilateral case, since then . The area can also be expressed in terms of the bimedians , and the diagonals , : In fact, any three of the four values , , , and suffice for determination of the area, since in any quadrilateral the four values are related by The corresponding expressions are: if the lengths of two bimedians and one diagonal are given, and if the lengths of two diagonals and one bimedian are given. Vector formulas The area of a quadrilateral can be calculated using vectors. Let vectors and form the diagonals from to and from to . The area of the quadrilateral is then which is half the magnitude of the cross product of vectors and . In two-dimensional Euclidean space, expressing vector as a free vector in Cartesian space equal to and as , this can be rewritten as: Diagonals Properties of the diagonals in quadrilaterals In the following table it is listed if the diagonals in some of the most basic quadrilaterals bisect each other, if their diagonals are perpendicular, and if their diagonals have equal length. The list applies to the most general cases, and excludes named subsets. Note 1: The most general trapezoids and isosceles trapezoids do not have perpendicular diagonals, but there are infinite numbers of (non-similar) trapezoids and isosceles