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In Euclidean geometry, a parallelogram is a
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
(non- self-intersecting) quadrilateral with two pairs of
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped. The etymology (in Greek παραλληλ-όγραμμον, ''parallēl-ógrammon'', a shape "of parallel lines") reflects the definition.


Special cases

*
Rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...
– A parallelogram with four angles of equal size (right angles). * Rhombus – A parallelogram with four sides of equal length. Any parallelogram that is neither a rectangle nor a rhombus was traditionally called a rhomboid but this term is not used in modern mathematics. * Square – A parallelogram with four sides of equal length and angles of equal size (right angles).


Characterizations

A
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
(non-self-intersecting) quadrilateral is a parallelogram
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
any one of the following statements is true: *Two pairs of opposite sides are parallel (by definition). *Two pairs of opposite sides are equal in length. *Two pairs of opposite angles are equal in measure. *The diagonals bisect each other. *One pair of opposite sides is
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
and equal in length. *
Adjacent angles In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
are supplementary. *Each diagonal divides the quadrilateral into two congruent
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
s. *The sum of the squares of the sides equals the sum of the squares of the diagonals. (This is the parallelogram law.) *It has
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...
of order 2. *The sum of the distances from any interior point to the sides is independent of the location of the point. (This is an extension of Viviani's theorem.) *There is a point ''X'' in the plane of the quadrilateral with the property that every straight line through ''X'' divides the quadrilateral into two regions of equal area. Thus all parallelograms have all the properties listed above, and
conversely In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...
, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram.


Other properties

*Opposite sides of a parallelogram are parallel (by definition) and so will never intersect. *The area of a parallelogram is twice the area of a triangle created by one of its diagonals. *The area of a parallelogram is also equal to the magnitude of the vector cross product of two
adjacent Adjacent or adjacency may refer to: * Adjacent (graph theory), two vertices that are the endpoints of an edge in a graph * Adjacent (music), a conjunct step to a note which is next in the scale See also * Adjacent angles, two angles that share ...
sides. *Any line through the midpoint of a parallelogram bisects the area. *Any non-degenerate affine transformation takes a parallelogram to another parallelogram. *A parallelogram has
rotational symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which ...
of order 2 (through 180°) (or order 4 if a square). If it also has exactly two lines of
reflectional symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D th ...
then it must be a rhombus or an oblong (a non-square rectangle). If it has four lines of reflectional symmetry, it is a square. *The perimeter of a parallelogram is 2(''a'' + ''b'') where ''a'' and ''b'' are the lengths of adjacent sides. *Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area. *The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square.Weisstein, Eric W. "Parallelogram." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Parallelogram.html *If two lines parallel to sides of a parallelogram are constructed concurrent to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area. *The diagonals of a parallelogram divide it into four triangles of equal area.


Area formula

All of the area formulas for general convex quadrilaterals apply to parallelograms. Further formulas are specific to parallelograms: A parallelogram with base ''b'' and height ''h'' can be divided into a trapezoid and a right triangle, and rearranged into a
rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...
, as shown in the figure to the left. This means that the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of a parallelogram is the same as that of a rectangle with the same base and height: :K = bh. The base × height area formula can also be derived using the figure to the right. The area ''K'' of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is :K_\text = (B+A) \times H\, and the area of a single orange triangle is :K_\text = \frac \times H. \, Therefore, the area of the parallelogram is :K = K_\text - 2 \times K_\text = ( (B+A) \times H) - ( A \times H) = B \times H. Another area formula, for two sides ''B'' and ''C'' and angle θ, is :K = B \cdot C \cdot \sin \theta.\, The area of a parallelogram with sides ''B'' and ''C'' (''B'' ≠ ''C'') and angle \gamma at the intersection of the diagonals is given byMitchell, Douglas W., "The area of a quadrilateral", ''Mathematical Gazette'', July 2009. :K = \frac \cdot \left, B^2 - C^2 \. When the parallelogram is specified from the lengths ''B'' and ''C'' of two adjacent sides together with the length ''D''1 of either diagonal, then the area can be found from Heron's formula. Specifically it is :K=2\sqrt where S=(B+C+D_1)/2 and the leading factor 2 comes from the fact that the chosen diagonal divides the parallelogram into ''two'' congruent triangles.


Area in terms of Cartesian coordinates of vertices

Let vectors \mathbf,\mathbf\in\R^2 and let V = \begin a_1 & a_2 \\ b_1 & b_2 \end \in\R^ denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to , \det(V), = , a_1b_2 - a_2b_1, \,. Let vectors \mathbf,\mathbf\in\R^n and let V = \begin a_1 & a_2 & \dots & a_n \\ b_1 & b_2 & \dots & b_n \end \in\R^. Then the area of the parallelogram generated by a and b is equal to \sqrt. Let points a,b,c\in\R^2. Then the area of the parallelogram with vertices at ''a'', ''b'' and ''c'' is equivalent to the absolute value of the determinant of a matrix built using ''a'', ''b'' and ''c'' as rows with the last column padded using ones as follows: :K = \left, \,\det \begin a_1 & a_2 & 1 \\ b_1 & b_2 & 1 \\ c_1 & c_2 & 1 \end \.


Proof that diagonals bisect each other

To prove that the diagonals of a parallelogram bisect each other, we will use congruent
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
s: :\angle ABE \cong \angle CDE ''(alternate interior angles are equal in measure)'' :\angle BAE \cong \angle DCE ''(alternate interior angles are equal in measure)''. (since these are angles that a transversal makes with parallel lines ''AB'' and ''DC''). Also, side ''AB'' is equal in length to side ''DC'', since opposite sides of a parallelogram are equal in length. Therefore, triangles ''ABE'' and ''CDE'' are congruent (ASA postulate, ''two corresponding angles and the included side''). Therefore, :AE = CE :BE = DE. Since the diagonals ''AC'' and ''BD'' divide each other into segments of equal length, the diagonals bisect each other. Separately, since the diagonals ''AC'' and ''BD'' bisect each other at point ''E'', point ''E'' is the midpoint of each diagonal.


Lattice of parallelograms

Parallelograms can tile the plane by translation. If edges are equal, or angles are right, the symmetry of the lattice is higher. These represent the four Bravais lattices in 2 dimensions.


Parallelograms arising from other figures


Automedian triangle

An
automedian triangle In plane geometry, an automedian triangle is a triangle in which the lengths of the three medians (the line segments connecting each vertex to the midpoint of the opposite side) are proportional to the lengths of the three sides, in a different or ...
is one whose
medians The Medes (Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, the ...
are in the same proportions as its sides (though in a different order). If ''ABC'' is an automedian triangle in which vertex ''A'' stands opposite the side ''a'', ''G'' is the centroid (where the three medians of ''ABC'' intersect), and ''AL'' is one of the extended medians of ''ABC'' with ''L'' lying on the circumcircle of ''ABC'', then ''BGCL'' is a parallelogram.


Varignon parallelogram

The midpoints of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its Varignon parallelogram. If the quadrilateral is convex or
concave Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon, a polygon which is not convex * Concave set In geometry, a subset o ...
(that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral.


Tangent parallelogram of an ellipse

For an ellipse, two diameters are said to be conjugate if and only if the tangent line to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding tangent parallelogram, sometimes called a bounding parallelogram, formed by the tangent lines to the ellipse at the four endpoints of the conjugate diameters. All tangent parallelograms for a given ellipse have the same area. It is possible to reconstruct an ellipse from any pair of conjugate diameters, or from any tangent parallelogram.


Faces of a parallelepiped

A parallelepiped is a three-dimensional figure whose six faces are parallelograms.


See also

*
Fundamental parallelogram (disambiguation) Fundamental parallelogram may mean: * Fundamental pair of periods In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. This type of lattice is the underlying object with w ...
*
Antiparallelogram In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, sides in the ...
*
Levi-Civita parallelogramoid In the mathematical field of differential geometry, the Levi-Civita parallelogramoid is a quadrilateral in a curved space whose construction generalizes that of a parallelogram in the Euclidean plane. It is named for its discoverer, Tullio Levi-C ...


References


External links


Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
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Interactive Parallelogram --sides, angles and slopeArea of Parallelogram
at cut-the-knot
Equilateral Triangles On Sides of a Parallelogram
at cut-the-knot
Definition and properties of a parallelogram
with animated applet

interactive applet {{Polygons Types of quadrilaterals Elementary shapes