this is about circles

1. Circle
From Wikipedia,the free encyclopedia
This article is about the shape and mathematical concept. For other uses, see Circle (disambiguation).
"360 degrees" redirects here. For other uses, see 360 degrees (disambiguation).
Circle
A circle (black) which is measured by its circumference (C), diameter
(D) in cyan, and radius (R) in red; its centre (O) is in magenta.
A circle is a simple closed shape in Euclidean geometry. It is the set of all points in a plane that are at a given
distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its
distance from a given point is constant. The distance between any of the points and the centre is called
the radius.
A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In
everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to
the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole
figure is called a disc.
A circle may also be defined as a special kind of ellipse in which the two foci are coincident and
the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared,
using calculus of variations.
Contents
[hide]
 1Terminology
 2History
 3Analytic results
o 3.1Length of circumference
o 3.2Area enclosed
o 3.3Equations
o 3.4Tangent lines

2.  4Properties
o 4.1Chord
o 4.2Tangent
o 4.3Theorems
o 4.4Inscribed angles
o 4.5Sagitta
 5Compass and straightedge constructions
o 5.1Construct a circle with a given diameter
o 5.2Construct a circle through 3 noncollinear points
 6Circle of Apollonius
o 6.1Cross-ratios
o 6.2Generalised circles
 7Circles inscribed in or circumscribed about other figures
 8Circle as limiting case of other figures
 9Squaring the circle
 10See also
o 10.1Specially named circles
 11References
 12Further reading
 13External links
A circle is a plane figure bounded by one line, and such that all right lines drawn from a certain point within it
to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.
— Euclid. Elements Book I.[1]:4
Terminology[edit]
 Annulus: the ring-shaped object, the region bounded by two concentric circles.
 Arc: any connected part of the circle.
 Centre: the point equidistant from the points on the circle.
 Chord: a line segment whose endpoints lie on the circle.
 Circumference: the length of one circuit along the circle, or the distance around the circle.
 Diameter: a line segment whose endpoints lie on the circle and which passes through the centre; or the
length of such a line segment, which is the largest distance between any two points on the circle. It is a
special case of a chord, namely the longest chord, and it is twice the radius.
 Disc: the region of the plane bounded by a circle
 Lens: the intersection of two discs
 Passant: a coplanar straight line that does not touch the circle.
 Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such
a segment, which is half a diameter.
 Sector:a region bounded by two radii and an arc lying between the radii.
 Segment: a region, not containing the centre, bounded by a chord and an arc lying between the chord's
endpoints.
 Secant: an extended chord, a coplanar straight line cutting the circle at two points.
 Semicircle: an arc that extends from one of a diameter's endpoints to the other. In non-technical common
usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a
half-disc. A half-disc is a special case of a segment, namely the largest one.
 Tangent: a coplanar straight line that touches the circle at a single point.

3. Chord,secant,tangent,radius,anddiameter
Arc,sector,and segment
History[edit]
The compass in this 13th-century manuscript is a symbolofGod's act of Creation.Notice also the circularshape of
the halo.
The word circle derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric
Greek κρίκος (krikos), meaning "hoop" or "ring".[2]
The origins of the words circus and circuit are closely
related.

4. Circular piece ofsilk with Mongolimages
Circles in an old Arabicastronomicaldrawing.
The circle has been known since before the beginning of recorded history. Natural circles would have been
observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle
shape in the sand. The circle is the basis for the wheel, which, with related inventions such asgears, makes
much of modern machinery possible. In mathematics, the study of the circle has helped inspire the
development of geometry, astronomy, and calculus.
Early science, particularly geometry and astrology and astronomy, was connected to the divine for
mostmedieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that
could be found in circles.[3][4]
Some highlights in the history of the circle are:
 1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds
to 256/81 (3.16049...) as an approximate value of π.[5]
TughrulTowerfrominside

5.  300 BCE – Book 3 of Euclid's Elements deals with the properties of circles.
 In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the
perfect circle, and how it is different from any drawing, words, definition or explanation.
 1880 CE – Lindemann proves that π is transcendental, effectively settling the millennia-old problem
ofsquaring the circle.[6]
Analytic results[edit]
Lengthof circumference[edit]
Further information: Circumference
The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to
3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:
Area enclosed[edit]
Area enclosedby a circle = π × area ofthe shadedsquare
Main article: Area of a circle
As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of
a triangle whose base has the length of the circle's circumference and whose height equals the circle's
radius,[7]
which comes to π multiplied by the radius squared:
Equivalently, denoting diameter by d,
that is, approximately 79% of the circumscribing square (whose side is of length d).
The circle is the plane curve enclosing the maximum area for a given arc length. This relates the
circle to a problem in the calculus of variations, namely the isoperimetric inequality.
Equations[edit]
Cartesian coordinates[edit]

6. Circle of radius r= 1, centre (a,b) = (1.2, −0.5)
In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the
set of all points (x, y) such that
This equation, known as the Equation of the Circle, follows from the Pythagorean
theorem applied to any point on the circle: as shown in the diagram to the right, the radius is
the hypotenuse of a right-angled triangle whose other sides are of length |x − a| and |y − b|. If
the circle is centred at the origin (0, 0), then the equation simplifies to
The equation can be written in parametric form using the trigonometric functions sine
and cosine as
where t is a parametric variable in the range 0 to 2π, interpreted geometrically
as the angle that the ray from (a, b) to (x, y) makes with the positive x-axis.
An alternative parametrisation of the circle is:
In this parametrisation, the ratio of t to r can be interpreted
geometrically as the stereographic projection of the line passing
through the centre parallel to the x-axis (see Tangent half-angle
substitution). However, this parametrisation works only if t is made to
range not only through all reals but also to a point at infinity;
otherwise, the bottom-most point of the circle would be omitted.
In homogeneous coordinates each conic section with the equation of a
circle has the form

7. It can be proven that a conic section is a circle exactly when it
contains (when extended to the complex projective plane) the
points I(1: i: 0) and J(1: −i: 0). These points are called
the circular points at infinity.
Polar coordinates[edit]
In polar coordinates the equation of a circle is:
where a is the radius of the circle, is the polar
coordinate of a generic point on the circle, and is the
polar coordinate of the centre of the circle (i.e., r0 is the
distance from the origin to the centre of the circle, and φ is
the anticlockwise angle from the positive x-axis to the line
connecting the origin to the centre of the circle). For a circle
centred at the origin, i.e. r0 = 0, this reduces to simply r = a.
When r0 = a, or when the origin lies on the circle, the
equation becomes
In the general case, the equation can be solved for r,
giving
Note that without the ± sign, the equation would in
some cases describe only half a circle.
Complex plane[edit]
In the complex plane, a circle with a centre at c and radius (r) has the equation
. In parametric form this can be written .
The slightly generalised equation for real p, q and complex g is sometimes called
a generalised circle. This becomes the above equation for a circle with , since . Not all
generalised circles are actually circles: a generalised circle is either a (true) circle or a line.
Tangent lines[edit]
Main article: Tangent lines to circles
The tangent line through a point P on the circle is perpendicular to the diameter passing
through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is
perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1 − a)x + (y1 – b)y = c.
Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is

8. or
If y1 ≠ b then the slope of this line is
This can also be found using implicit differentiation.
When the centre of the circle is at the origin then the equation of the tangent line
becomes
and its slope is
Properties[edit]
 The circle is the shape with the largest area for a given length of
perimeter. (See Isoperimetric inequality.)
 The circle is a highly symmetric shape: every line through the centre
forms a line of reflection symmetry and it has rotational
symmetry around the centre for every angle. Itssymmetry group is
the orthogonal group O(2,R). The group of rotations alone is the circle
group T.
 All circles are similar.
 A circle's circumference and radius are proportional.
 The area enclosed and the square of its radius are proportional.
 The constants of proportionality are 2π and π, respectively.
 The circle which is centred at the origin with radius 1 is called the unit
circle.
 Thought of as a great circle of the unit sphere, it becomes
the Riemannian circle.
 Through any three points, not all on the same line, there lies a unique
circle. In Cartesian coordinates, it is possible to give explicit formulae
for the coordinates of the centre of the circle and the radius in terms of
the coordinates of the three given points. See circumcircle.
Chord[edit]
 Chords are equidistant from the centre of a circle if and only if they are
equal in length.

9.  The perpendicular bisector of a chord passes through the centre of a
circle; equivalent statements stemming from the uniqueness of the
perpendicular bisector are:
 A perpendicular line from the centre of a circle bisects the chord.
 The line segment through the centre bisecting a chord
is perpendicular to the chord.
 If a central angle and an inscribed angle of a circle are subtended by the
same chord and on the same side of the chord, then the central angle is
twice the inscribed angle.
 If two angles are inscribed on the same chord and on the same side of the
chord, then they are equal.
 If two angles are inscribed on the same chord and on opposite sides of
the chord, then they are supplementary.
 For a cyclic quadrilateral, the exterior angle is equal to the interior
opposite angle.
 An inscribed angle subtended by a diameter is a right angle (see Thales'
theorem).
Tangent[edit]
 A line drawn perpendicular to a radius through the end point of the
radius lying on the circle is a tangent to the circle.
 A line drawn perpendicular to a tangent through the point of contact with
a circle passes through the centre of the circle.
 Two tangents can always be drawn to a circle from any point outside the
circle, and these tangents are equal in length.
 If a tangent at A and a tangent at B intersect at the exterior point P, then
denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary.
 If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 2arc(AQ).
Theorems[edit]
Secant-secant theorem

10. See also: Power of a point
 The chord theorem states that if two chords, CD and EB, intersect at A,
then CA × DA = EA × BA.
 If a tangent from an external point D meets the circle at C and a secant from the external
point D meets the circle at G and Erespectively, then DC2
= DG × DE. (Tangent-secant
theorem.)
 If two secants, DG and DE, also cut the circle at H and F respectively, then DH × DG = DF × DE.
(Corollary of the tangent-secant theorem.)
 The angle between a tangent and chord is equal to one half the angle subtended at the centre of
the circle, on the opposite side of the chord (Tangent Chord Angle).
 If the angle subtended by the chord at the centre is 90 degrees then l = r √2, where l is the
length of the chord and r is the radius of the circle.
 If two secants are inscribed in the circle as shown at right, then the measurement of angle A is
equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is
the secant-secant theorem.
Inscribed angles[edit]
See also: Inscribed angle theorem
Inscribed angle theorem
An inscribed angle (examples are the blue and green angles in the figure) is exactly half the
corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are
equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle
that subtends a diameter is a right angle (since the central angle is 180 degrees).
Sagitta[edit]

11. The sagitta is the vertical segment.
 The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord,
between the midpoint of that chord and the arc of the circle.
 Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be
used to calculate the radius of the unique circle which will fit around the two lines:
Another proof of this result which relies only on two chord properties given above is as follows.
Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of
the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius,
the "missing" part of the diameter is (2r − x) in length. Using the fact that one part of one chord
times the other part is equal to the same product taken along a chord intersecting the first chord,
we find that (2r − x)x = (y / 2)2
. Solving for r, we find the required result.
Compass and straightedge constructions[edit]
There are many compass-and-straightedge constructions resulting in circles.
The simplest and most basic is the construction given the centre of the circle and a point on the
circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on
the circle and rotate the compass.
Construct a circle with a given diameter[edit]
 Construct the midpoint M of the diameter.
 Construct the circle with centre M passing through one of the endpoints of the diameter (it
will also pass through the other endpoint).
Construct a circle through 3 noncollinear points[edit]
 Name the points P, Q and R,
 Construct the perpendicular bisector of the segment PQ.
 Construct the perpendicular bisector of the segment PR.
 Label the point of intersection of these two perpendicular bisectors M. (They meet because
the points are not collinear).

12.  Construct the circle with centre M passing through one of the points P, Q or R (it will also
pass through the other two points).
Circle of Apollonius[edit]
See also: Circles of Apollonius
Apollonius' definition of a circle: d1 / d2constant
Apollonius of Perga showed that a circle may also be defined as the set of points in a plane
having a constant ratio (other than 1) of distances to two fixed foci, A and B.[11][12]
(The set of
points where the distances are equal is the perpendicular bisector of A and B, a line.) That circle
is sometimes said to be drawn about two points.
The proof is in two parts. First, one must prove that, given two foci A and B and a ratio of
distances, any point P satisfying the ratio of distances must fall on a particular circle. Let C be
another point, also satisfying the ratio and lying on segment AB. By the angle bisector
theorem the line segment PC will bisect the interior angle APB, since the segments are similar:
Analogously, a line segment PD through some point D on AB extended bisects the
corresponding exterior angle BPQ where Q is onAP extended. Since the interior and exterior
angles sum to 180 degrees, the angle CPD is exactly 90 degrees, i.e., a right angle. The set
of points P such that angle CPD is a right angle forms a circle, of which CD is a diameter.
Second, see[13]:p.15
for a proof that every point on the indicated circle satisfies the given ratio.
Cross-ratios[edit]
A closely related property of circles involves the geometry of the cross-ratio of points in
the complex plane. If A, B, and C are as above, then the circle of Apollonius for these three
points is the collection of points P for which the absolute value of the cross-ratio is equal to
one:

13. Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio
[A,B;C,P] is on the unit circle in the complex plane.
Generalised circles[edit]
See also: Generalised circle
If C is the midpoint of the segment AB, then the collection of points P satisfying the
Apollonius condition
is not a circle, but rather a line.
Thus, if A, B, and C are given distinct points in the plane, then the locus of
points P satisfying the above equation is called a "generalised circle." It may either
be a true circle or a line. In this sense a line is a generalised circle of infinite radius.
Circles inscribed in or circumscribed about other
figures[edit]
In every triangle a unique circle, called the incircle, can be inscribed such that it
is tangent to each of the three sides of the triangle.[14]
About every triangle a unique circle, called the circumcircle, can be circumscribed
such that it goes through each of the triangle's three vertices.[15]
A tangential polygon, such as a tangential quadrilateral, is any convex
polygon within which a circle can be inscribed that is tangent to each side of the
polygon.[16]
A cyclic polygon is any convex polygon about which a circle can be circumscribed,
passing through each vertex. A well-studied example is the cyclic quadrilateral.
A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on
a smaller circle that rolls within and tangent to the given circle.
Circle as limiting case of other figures[edit]
The circle can be viewed as a limiting case of each of various other figures:
 A Cartesian oval is a set of points such that a weighted sum of the distances
from any of its points to two fixed points (foci) is a constant. An ellipse is the
case in which the weights are equal. A circle is an ellipse with an eccentricity of
zero, meaning that the two foci coincide with each other as the centre of the
circle. A circle is also a different special case of a Cartesian oval in which one of

14. the weights is zeroA superellipse has an equation of the form for
positive a, b, and n. A supercircle has b = a. A circle is the special case of a
supercircle in which n = 2.
 A Cassini oval is a set of points such that the product of the distances from any of its points to
two fixed points is a constant. When the two fixed points coincide, a circle results.
 A curve of constant width is a figure whose width, defined as the perpendicular distance
between two distinct parallel lines each intersecting its boundary in a single point, is the same
regardless of the direction of those two parallel lines. The circle is the simplest example of this
type of figure.
Squaring the circle[edit]
Squaring the circle is the problem, proposed by ancient geometers, of constructing a square with the
same area as a given circle by using only a finite number of steps withcompass and straightedge.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass
theorem which proves that pi (π) is a transcendental number, rather than analgebraic irrational
number; that is, it is not the root of any polynomial with rational coefficients.
 Interactive Java applets for the properties of and elementary constructions involving circles.
 Interactive Standard Form Equation of Circle Click and drag points to see standard form
equation in action
 Munching on Circles at cut-the-knot
Quadrilateral
From Wikipedia, the free encyclopedia
This article is about four-sided mathematical shapes. For other uses, see Quadrilateral
(disambiguation).
Quadrilateral

15. Some types of quadrilaterals
Edges and vertices 4
Schläfli symbol {4} (for square)
Area various methods;
see below
Internal angle(degrees) 90° (for square and rectangle)
In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and
four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and
sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on.
The origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus,
meaning "side".
Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed.
Simple quadrilaterals are eitherconvexor concave.
The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is
This is a special case of the n-gon interior angle sum formula (n − 2) × 180°.
All non-self-crossing quadrilaterals tile the plane by repeated rotation around the midpoints of
their edges.
Contents
[hide]
 1Simple quadrilaterals

16. o 1.1Convex quadrilaterals
o 1.2Concave quadrilaterals
 2Complex quadrilaterals
 3Special line segments
 4Area of a convex quadrilateral
o 4.1Trigonometric formulas
o 4.2Non-trigonometric formulas
o 4.3Vector formulas
 5Diagonals
o 5.1Properties of the diagonals in some quadrilaterals
o 5.2Lengths of the diagonals
o 5.3Generalizations of the parallelogram law and Ptolemy's theorem
o 5.4Other metric relations
 6Angle bisectors
 7Bimedians
 8Trigonometric identities
 9Inequalities
o 9.1Area
o 9.2Diagonals and bimedians
o 9.3Sides
 10Maximum and minimum properties
 11Remarkable points and lines in a convex quadrilateral
 12Other properties of convex quadrilaterals
 13Taxonomy
 14Skew quadrilaterals
 15See also
 16References
 17External links
Simple quadrilaterals[edit]
Any quadrilateral that is not self-intersecting is a simple quadrilateral.
Convex quadrilaterals[edit]

17. Euler diagram of some types of simple quadrilaterals. (UK) denotes British English and (US) denotes
American English.
In a convex quadrilateral, all interior angles are less than 180° and the two diagonals both lie
inside the quadrilateral.
 Irregular quadrilateral (British English) or trapezium (North American English): no sides are
parallel. (In British English thiswas once called a trapezoid.)
 Trapezium (UK) or trapezoid (US): at least one pair of opposite sides are parallel.
 Isosceles trapezium (UK) or isosceles trapezoid (US): one pair of opposite sides are parallel
and the base angles are equal in measure. Alternative definitions are a quadrilateral with an
axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal
length.
 Parallelogram: a quadrilateral with two pairs of parallel sides. Equivalent conditions are that
opposite sides are of equal length; that opposite angles are equal; or that the diagonals
bisect each other. Parallelograms also include the square, rectangle, rhombus and
rhomboid.
 Rhombus or rhomb: all four sides are of equal length. An equivalent condition is that the
diagonals perpendicularly bisect each other. Informally: "a pushed-over square" (but strictly
including a square too).
 Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles
are oblique (not right angles). A parallelogram which is not a rhombus. Informally: "a
pushed-over oblong" (but strictly including an oblong too).[1]
 Rectangle: all four angles are right angles. An equivalent condition is that the diagonals
bisect each other and are equal in length. Informally: "a box or oblong" (including a square).
 Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four
angles are right angles. An equivalent condition is that opposite sides are parallel (a square
is a parallelogram), that the diagonals perpendicularly bisect each other, and are of equal
length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (four
equal sides and four equal angles).
 Oblong: a term sometimes used to denote a rectangle which has unequal adjacent sides (i.e.
a rectangle that is not a square).[2]

18.  Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides
the kite into congruent triangles, and so the angles between the two pairs of equal sides are
equal in measure. It also implies that the diagonals are perpendicular.
 Tangential quadrilateral: the four sides are tangents to an inscribed circle. A convex
quadrilateral is tangential if and only if opposite sides have equal sums.
 Tangential trapezoid: a trapezoid where the four sides are tangents to an inscribed circle.
 Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is
cyclic if and only if opposite angles sum to 180°.
 Right kite: a kite with two opposite right angles. It is a type of cyclic quadrilateral.
 Bicentric quadrilateral: it is both tangential and cyclic.
 Orthodiagonal quadrilateral: the diagonals cross at right angles.
 Equidiagonal quadrilateral: the diagonals are of equal length.
 Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.
 An equilic quadrilateral has two opposite equal sides that, when extended, meet at 60°.
 A Watt quadrilateral is a quadrilateral with a pair of opposite sides of equal length.[3]
 A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter
of a square.[4]
 A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the
circumcircle.[5]
Concave quadrilaterals[edit]
In a concave quadrilateral, one interior angle is bigger than 180° and one of the two diagonals
lies outside the quadrilateral.
 A dart (or arrowhead) is a concave quadrilateral with bilateral symmetry like a kite, but one
interior angle is reflex.
Complex quadrilaterals[edit]

19. An antiparallelogram
A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed
quadrilateral, butterfly quadrilateral or bow-tiequadrilateral. In a crossed quadrilateral, the
four "interior" angles on either side of the crossing (two acute and two reflex, all on the left or all
on the right as the figure is traced out) add up to 720°.[6]
 Antiparallelogram: a crossed quadrilaterals in which (like a parallelogram) each pair of
nonadjacent sides have equal lengths.
 Crossed rectangle: an antiparallelogram whose sides are two opposite sides and the two
diagonals of a rectangle, hence having one pair of opposite sides parallel.
 Crossed square: a special case of a crossed rectangle where two of the sides intersect at
right angles.
Special line segments[edit]
The two diagonals of a convex quadrilateral are the line segments that connect opposite
vertices.
The two bimedians of a convexquadrilateral are the line segments that connect the midpoints
of opposite sides.[7]
They intersect at the "vertex centroid" of the quadrilateral (seeRemarkable
points below).
The four maltitudes of a convex quadrilateral are the perpendiculars to a side through the
midpoint of the opposite side.[8]
Area of a convex quadrilateral[edit]
There are various general formulas for the area K of a convex quadrilateral ABCD with sides a =
AB, b = BC, c = CD and d = DA.
Trigonometric formulas[edit]
The area can be expressed in trigonometric terms as
where the lengths of the diagonals are p and q and the angle between them is θ.[9]
In the
case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces
to since θ is 90°.
The area can be also expressed in terms of bimedians as[10]

20. where the lengths of the bimedians are m and n and the angle between them is φ.
Bretschneider's formula[11]
expresses the area in terms of the sides and two opposite
angles:
where the sides in sequence are a, b, c, d, where s is the semiperimeter,
and A and C are two (in fact, any two) opposite angles. This reduces
to Brahmagupta's formula for the area of a cyclic quadrilateral when A+C = 180°.
Another area formula in terms of the sides and angles, with angle C being between
sides b and c, and A being between sides a and d, 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 θ of the
diagonals, so long as this angle is not 90°:[12]
In the case of a parallelogram, the latter formula becomes
Another area formula including the sides a, b, c, d is[10]
where x is the distance between the midpoints of the diagonals
and φ is the angle between the bimedians.
The last trigonometric area formula including the
sides a, b, c, d and the angle α between a and b is:[citation needed]
which can also be used for the area of a concave
quadrilateral (having the concave part opposite to angle α)
just changing the first sign + to - .
Non-trigonometric formulas[edit]
The following two formulas express the area in terms of the
sides a, b, c, d, the semiperimeter s, and the diagonals p, q:
[13]

21. [14]
The first reduces to Brahmagupta's formula in the
cyclic quadrilateral case, since then pq = ac + bd.
The area can also be expressed in terms of the
bimedians m, n and the diagonals p, q:
[15]
[16]:Thm. 7
In fact, any three of the four values m, n, p,
and q suffice for determination of the area,
since in any quadrilateral the four values
are related by [17]:p. 126
The
corresponding expressions are:[citation needed]
if the lengths of two bimedians and one
diagonal are given, and[citation needed]
if the lengths of two diagonals and
one bimedian are given.
Vector formulas[edit]
The area of a
quadrilateral ABCD can be
calculated using vectors. Let
vectors AC and BD form the
diagonals from A to C and
from B to D. The area of the
quadrilateral is then
which is half the magnitude of
the cross product of
vectors AC and BD. In two-
dimensional Euclidean space,
expressing vector AC as
a free vector in Cartesian
space equal to (x1,y1)
and BD as (x2,y2), this can be
rewritten as:

22. Diagonals[edit]
Properties of the diagonals in some quadrilaterals[edit]
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.[18]
The list applies
to the most general cases, and excludes named subsets.
al Bisecting diagonals Perpendicular diagonals Equal diagonals
No See note 1 No
apezoid No See note 1 Yes
am Yes No No
See note 2 Yes See note 2
Yes No Yes
Yes Yes No
Yes Yes Yes
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 trapezoids that do have
perpendicular diagonals and are not any other named quadrilateral.
Note 2: In a kite, one diagonal bisects the other. The most general kite has unequal diagonals, but
there is an infinite number of (non-similar) kites in which the diagonals are equal in length (and the
kites are not any other named quadrilateral).
Lengths of the diagonals[edit]
The lengths of the diagonals in a convex quadrilateral ABCD can be calculated using the law of
cosines on each triangle formed by one diagonal and two sides of the quadrilateral. Thus
and

23. Other, more symmetric formulas for the lengths of the diagonals, are[19]
and
Generalizations of the parallelogram law and Ptolemy's
theorem[edit]
In any convex quadrilateral ABCD, the sum of the squares of the four sides is equal
to the sum of the squares of the two diagonals plus four times the square of the line
segment connecting the midpoints of the diagonals. Thus
where x is the distance between the midpoints of the diagonals.[17]:p.126
This is
sometimes known as Euler's quadrilateral theorem and is a generalization of
the parallelogram law.
The German mathematician Carl Anton Bretschneider derived in 1842 the
following generalization of Ptolemy's theorem, regarding the product of the
diagonals in a convex quadrilateral[20]
This relation can be considered to be a law of cosines for a quadrilateral. In
a cyclic quadrilateral, where A + C = 180°, it reduces to pq = ac + bd. Since
cos (A + C) ≥ −1, it also gives a proof of Ptolemy's inequality.