Parallelogram

Parallelogram
Parallelogram
	This parallelogram is a rhomboid as it has no right angles and unequal sides.
Type	quadrilateral, trapezium
Edges and vertices	4
Symmetry group	C2, [2]+,
Area	b × h (base × height);; ab sin θ (product of adjacent sides and sine of the vertex angle determined by them)
Properties	convex

Last updated February 09, 2024

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel 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.

Special cases
Characterizations
Other properties
Area formula
From vertex coordinates
Proof that diagonals bisect each other
Lattice of parallelograms
Parallelograms arising from other figures
Automedian triangle
Varignon parallelogram
Tangent parallelogram of an ellipse
Faces of a parallelepiped
See also
References
External links

By comparison, a quadrilateral with at least 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 – 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.^[1]
Square – A parallelogram with four sides of equal length and angles of equal size (right angles).

Characterizations

A simple (non-self-intersecting) quadrilateral is a parallelogram if and only if any one of the following statements is true:^[2]^[3]

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 and equal in length.
Adjacent angles are supplementary.
Each diagonal divides the quadrilateral into two congruent triangles.
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 of order 2.
The sum of the distances from any interior point to the sides is independent of the location of the point.^[4] (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.^[5]

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 of two adjacent sides.
Any line through the midpoint of a parallelogram bisects the area.^[6]
Any non-degenerate affine transformation takes a parallelogram to another parallelogram.
A parallelogram has rotational symmetry of order 2 (through 180°) (or order 4 if a square). If it also has exactly two lines of reflectional symmetry 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.^[7]
The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square.^[8]
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.^[8]
The diagonals of a parallelogram divide it into four triangles of equal area.

Area formula

A parallelogram can be rearranged into a rectangle with the same area. ParallelogramArea.svg — A parallelogram can be rearranged into a rectangle with the same area.

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, as shown in the figure to the left. This means that the area 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{rect}}=(B+A)\times H\,

and the area of a single triangle is

K_{\text{tri}}={\frac {A}{2}}\times H.\,

Therefore, the area of the parallelogram is

K=K_{\text{rect}}-2\times K_{\text{tri}}=((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 by^[9]

K={\frac {|\tan \gamma |}{2}}\cdot \left|B^{2}-C^{2}\right|.

When the parallelogram is specified from the lengths B and C of two adjacent sides together with the length D₁ of either diagonal, then the area can be found from Heron's formula. Specifically it is

K=2{\sqrt {S(S-B)(S-C)(S-D_{1})}}

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.

From vertex coordinates

Let vectors $\mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{2}$ and let $V={\begin{bmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{bmatrix}}\in \mathbb {R} ^{2\times 2}$ denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to $|\det(V)|=|a_{1}b_{2}-a_{2}b_{1}|\,$ .

Let vectors $\mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}$ and let $V={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\\b_{1}&b_{2}&\dots &b_{n}\end{bmatrix}}\in \mathbb {R} ^{2\times n}$ . Then the area of the parallelogram generated by a and b is equal to ${\sqrt {\det(VV^{\mathrm {T} })}}$ .

Let points $a,b,c\in \mathbb {R} ^{2}$ . Then the signed area of the parallelogram with vertices at a, b and c is equivalent to the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:

K=\left|{\begin{matrix}a_{1}&a_{2}&1\\b_{1}&b_{2}&1\\c_{1}&c_{2}&1\end{matrix}}\right|.

Proof that diagonals bisect each other

Parallelogram ABCD Parallelogram1.svg — Parallelogram ABCD

To prove that the diagonals of a parallelogram bisect each other, we will use congruent triangles:

\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.

Lattices
Form	Square	Rectangle	Rhombus	Rhomboid
System	Square (tetragonal)	Rectangular (orthorhombic)	Centered rectangular (orthorhombic)	Oblique (monoclinic)
Constraints	α=90°, a=b	α=90°	a=b	None
Symmetry	p4m, [4,4], order 8n	pmm, [∞,2,∞], order 4n		p1, [∞⁺,2,∞⁺], order 2n
Form

Parallelograms arising from other figures

Automedian triangle

An automedian triangle is one whose medians 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

Varignon's theorem holds that 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 (that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral.

Proof without words (see figure):

An arbitrary quadrilateral and its diagonals.
Bases of similar triangles are parallel to the blue diagonal.
Ditto for the red diagonal.
The base pairs form a parallelogram with half the area of the quadrilateral, A_q, as the sum of the areas of the four large triangles, A_l is 2 A_q (each of the two pairs reconstructs the quadrilateral) while that of the small triangles, A_s is a quarter of A_l (half linear dimensions yields quarter area), and the area of the parallelogram is A_q minus A_s.

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.

Related Research Articles

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, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices $,, and is sometimes denoted as .$

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex.

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 ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.

<span class="mw-page-title-main">Perpendicular</span> Relationship between two lines that meet at a right angle (90 degrees)

In geometry, two geometric objects are perpendicular if their intersection forms right angles at the point of intersection called a foot. The condition of perpendicularity may be represented graphically using the perpendicular symbol, ⟂. Perpendicular intersections can happen between two lines, between a line and a plane, and between two planes.

<span class="mw-page-title-main">Kite (geometry)</span> Quadrilateral symmetric across a diagonal

In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

<span class="mw-page-title-main">Bisection</span> Division of something into two equal or congruent parts

In geometry, bisection is the division of something into two equal or congruent parts. Usually it involves a bisecting line, also called a bisector. The most often considered types of bisectors are the segment bisector, a line that passes through the midpoint of a given segment, and the angle bisector, a line that passes through the apex of an angle . In three-dimensional space, bisection is usually done by a bisecting plane, also called the bisector.

<span class="mw-page-title-main">Rhombus</span> Quadrilateral in which all sides have the same length

In plane Euclidean geometry, a rhombus 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 rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

<span class="mw-page-title-main">Trapezoid</span> Convex quadrilateral with at least one pair of parallel sides

In geometry, a trapezoid in North American English, or trapezium in British English, is a quadrilateral that has at least one pair of parallel sides.

In geometry, Thales's theorem states that if $A$ , $B$ , and $C$ are distinct points on a circle where the line $AC$ is a diameter, the angle $∠ ABC$ is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.

<span class="mw-page-title-main">Midpoint</span> Point on a line segment which is equidistant from both endpoints

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.

In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of equal measure, or as a trapezoid whose diagonals have equal length. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides are parallel, and the two other sides are of equal length, and the diagonals have equal length. The base angles of an isosceles trapezoid are equal in measure.

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted $ABCD .$

In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point.

<span class="mw-page-title-main">Antiparallelogram</span> Polygon with four crossed edges of two lengths

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, each pair of sides is antiparallel with respect to the other, with sides in the longer pair crossing each other as in a scissors mechanism. Whereas a parallelogram's opposite angles are equal and oriented the same way, an antiparallelogram's are equal but oppositely oriented. Antiparallelograms are also called contraparallelograms or crossed parallelograms.

In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.

In Euclidean geometry, Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon parallelogram. It is named after Pierre Varignon, whose proof was published posthumously in 1731.

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.

References

↑ "CIMT - Page no longer available at Plymouth University servers" (PDF). www.cimt.plymouth.ac.uk. Archived from the original (PDF) on 2014-05-14.
↑ Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, pp. 51-52.
↑ Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.
↑ Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", The College Mathematics Journal 37(5), 2006, pp. 390–391.
↑ Problem 5, 2006 British Mathematical Olympiad, .
↑ Dunn, J.A., and J.E. Pretty, "Halving a triangle", Mathematical Gazette 56, May 1972, p. 105.
↑ Weisstein, Eric W. "Triangle Circumscribing". Wolfram Math World.
1 2 Weisstein, Eric W. "Parallelogram." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Parallelogram.html
↑ Mitchell, Douglas W., "The area of a quadrilateral", Mathematical Gazette, July 2009.

External links

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[1] "CIMT - Page no longer available at Plymouth University servers" (PDF). www.cimt.plymouth.ac.uk. Archived from the original (PDF) on 2014-05-14.

[2] Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry, Mathematical Association of America, 2010, pp. 51-52.

[3] Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.

[4] Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", The College Mathematics Journal 37(5), 2006, pp. 390–391.

[5] Problem 5, 2006 British Mathematical Olympiad, .

[6] Dunn, J.A., and J.E. Pretty, "Halving a triangle", Mathematical Gazette 56, May 1972, p. 105.

[7] Weisstein, Eric W. "Triangle Circumscribing". Wolfram Math World.

[Weisstein-8] 1 2 Weisstein, Eric W. "Parallelogram." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Parallelogram.html

[9] Mitchell, Douglas W., "The area of a quadrilateral", Mathematical Gazette, July 2009.

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