In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, 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
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
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 of ...
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
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
''If a line segmen ...
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
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a convex quadrilateral in Eu ...
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 – A parallelogram with four angles of equal size (right angles).
*
Rhombus
In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...
– A parallelogram with four sides of equal length. Any parallelogram that is neither a rectangle nor a rhombus was traditionally called a
rhomboid
Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled.
A parallelogram with sides of equal length (equilateral) is a rhombus but not a rhomboi ...
but this term is not used in modern mathematics.
*
Square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
– 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
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
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 b ...
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
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s 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 of ...
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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
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
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
s of the sides equals the sum of the squares of the diagonals. (This is the
parallelogram law
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the s ...
.)
*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 i ...
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
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in various ma ...
.)
*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, 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
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
of two
adjacent 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 i ...
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 ther ...
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
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
.
*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
Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to:
Law
* Concurrence, in jurisprudence, the need to prove both ''actus reus'' and ''mens rea''
* Concurring opinion (also called a "concurrence"), a ...
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
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a convex quadrilateral in Eu ...
and a right triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
, and rearranged into a rectangle, 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 ope ...
of a parallelogram is the same as that of a rectangle with the same base and height:
:
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
:
and the area of a single orange triangle is
:
Therefore, the area of the parallelogram is
:
Another area formula, for two sides ''B'' and ''C'' and angle θ, is
:
The area of a parallelogram with sides ''B'' and ''C'' (''B'' ≠ ''C'') and angle at the intersection of the diagonals is given by[Mitchell, Douglas W., "The area of a quadrilateral", ''Mathematical Gazette'', July 2009.]
:
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
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is,
:A = \sqrt.
It is named after first-century ...
. Specifically it is
:
where 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 and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to .
Let vectors and let . Then the area of the parallelogram generated by a and b is equal to .
Let points . 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:
:
Proof that diagonals bisect each other
To prove that the diagonals of a parallelogram bisect each other, we will use congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
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:
: ''(alternate interior angles are equal in measure)''
: ''(alternate interior angles are equal in measure)''.
(since these are angles that a transversal makes with parallel lines
In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not touch each other or int ...
''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,
:
:
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
![varignon_parallelogram](https://upload.wikimedia.org/wikipedia/commons/b/b8/Varignon_parallelogram.svg)
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, th ...
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
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
(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 midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.
Formula
The midpoint of a segment in ''n''-dimen ...
s of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its Varignon parallelogram. If the quadrilateral is convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
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
* The concavity of a ...
(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
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
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
Reconstruction may refer to:
Politics, history, and sociology
*Reconstruction (law), the transfer of a company's (or several companies') business to a new company
*'' Perestroika'' (Russian for "reconstruction"), a late 20th century Soviet Unio ...
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
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
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 whi ...
*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 lo ...
* 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 ...
References
External links
Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
*
Interactive Parallelogram --sides, angles and slope
Area 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