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 Skip Navigation LinksMath Help > Geometry > Quadrilaterals

Classification of quadrilaterals

Depiction of a hierarchy of quadrilaterals, in which the green arrows represent additional properties.  For example, a parallelogram becomes a rhombus by making all four sides the same length.  Similar charts can be found in Mathwords, Wikipedia, and Dr.Math.
Figure Definition, followed by additional facts (theorems) Internet references
A closed plane figure consisting of four line segments.  A "crossed" (or "complex") quadrilateral has one pair of intersecting sides.  A "concave" quadrilateral, pictured here, has one interior angle greater than 180°.  The other possibility, described next, is "convex".  Bretschneider's Formula gives the area of a quadrilateral with sides of length a, b, c, d and opposite interior angles A and C.  The sum of the interior angles of a non-crossed quadrilateral 360.  In a crossed quadrilateral, the sum of the interior angles on one side of the crossing equals the sum of the interior angles on the other side of the crossing. Mathwords, Mathworld, Dr.Math, Wikipedia, Geometry Atlas, Math.com, Math is fun 
"convex" means every line segment connecting interior points is entirely contained within the interior.  Theorems: A case of Ramsey's Theorem tells us: Given any five points in a plane with no three collinear, four are the vertices of a convex quadrilateral. Mathwords, Dr.Math 
A quadrilateral with two pairs of adjacent equal sides.  (In some text, a kite need not be convex; in others concave kites are termed a "dart" or "arrowhead".)  Theorems: One set of opposite angles is equal, and that one diagonal perpendicularly bisects the other.  The bisecting diagonal forms an axis of symmetry, dividing the kite into two congruent triangles.  The bisected diagonal divides the kite into two isosceles triangles.  A convex kite is tangential (inscriptable).  A quadrilateral has bilateral symmetry iff it is either a kite or an isosceles trapezoid. Mathwords, Mathworld, Dr.Math, Wikipedia, Geometry Atlas 
A convex quadrilateral whose four vertices lie on a circumscribed circle.  Theorems: In a cyclic quadrilateral, opposite angles are supplementary.  The area of a cyclic quadrilateral is the maximum possible for any quadrilateral with the given side lengths.  Brahmagupta's formula gives the area of a cyclic quadrilateral, using only the side lengths.  Cyclic quadrilaterals that are also inscriptable are called "bicentric". Mathworld, Wikipedia, Geometry Atlas  
A convex quadrilateral with one pair of parallel sides.  Theorems: Two adjacent angles of a convex quadrilateral are supplementary iff it is a trapezoid.  The diagonals cut each other in proportion of the lengths of the parallel sides.  Mathwords, Mathworld, Dr.Math, Wikipedia, Geometry Atlas 
A kite which is also cyclic.  Theorems: A cyclic kite has a pair of opposite right angles.  Being a kite, it has a pair of congruent angles.  Being cyclic, these congruent angles must also be supplementary, so they are right angles.  
A trapezoid with two opposite sides parallel, the two other sides are of equal length. This implies that the two ends of each parallel side have equal angles, and that the diagonals are of equal length.  A quadrilateral has bilateral symmetry iff it is either a kite or an isosceles trapezoid. Mathwords, Mathworld, Wikipedia 
A convex quadrilateral in which both pairs of opposite sides are parallel.  This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other.  Each diagonal bisects the parallelogram into two congruent triangles.
Euclid showed if lines parallel to the sides are drawn through any point on a diagonal of a parallelogram, then the parallelograms not containing segments of that diagonal are equal in area (and conversely).
Varignon's Theorem: A parallelogram is formed by joining the midpoints of adjacent sides of a quadrilateral.  The center of Varignon's Parallelogram is the centroid of the vertices of the quadrilateral.
Wittenbauer's Theorem: A parallelogram is formed by dividing the sides of a quadrilateral into three equal parts, and connecting and extending adjacent points on either side of each vertex.  The center of Wittenbauer's parallelogram is the quadrilateral's centroid.
Mathwords, Mathworld, Wikipedia, Geometry Atlas 
A convex quadrilateral with four right angles.  This implies that opposite sides are parallel and of equal length, and the diagonals bisect each other and are equal in length. Mathworld, Wikipedia, Geometry Atlas 
A convex quadrilateral with all four sides of equal length.  This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other.  Its area is given by A=bh, where b is the base, and h is the height, or perpendicular distance between opposite sides.  A rhombus is tangential (inscriptable). Mathwords, Mathworld, Wikipedia, Geometry Atlas 
A convex quadrilateral with four sides of equal length, and four right angles.  This implies that opposite sides are parallel, and that the diagonals perpendicularly bisect each other and are of equal length.  Each diagonal bisects each pair of opposite angles. Mathwords, Mathworld, Wikipedia, Geometry Atlas 
Tangential quadrilateral
Inscriptable quadrilateral
A convex quadrilateral in which a circle can be inscribed, tangent to all four sides.  Theorems: The four angle bisectors meet (at the center of the inscribed circle) iff the figure is tangential.  Pairs of opposites sides sum to the same number, which is the semiperimeter of the quadrilateral.  Cyclic inscriptable quadrilaterals are also called "bicentric" Wikipedia 
A quadrilateral with just one pair of congruent opposite sides, and just one pair of congruent opposite angles is not necessarily a parallelogram.  The reason, as explained in Dr. Math, can be seen by drawing the shorter diagonal in the figure to the left, which divides the figure into two non-congruent triangles which nonetheless have congruent side-side-angle (SSA). Dr.Math 

Related pages in this website:

Geometry Glossary defines geometrical terms.

Bretschneider's Formula gives the area of a quadrilateral with sides of length a, b, c, d and opposite interior angles A and C.

Brahmagupta's formula gives the area of a cyclic quadrilateral, using only the side lengths.

Internet references

Quadrilaterals are referenced by Mathwords, Mathworld, Dr.Math, Wikipedia, Geometry Atlas, Math.com, Math is fun.

 

The webmaster and author of this Math Help site is Graeme McRae.