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 (e.g. pentagon ). Since "gon" means "angle", it is analogously called a quadrangle , or 4-angle. A quadrilateral with vertices A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} and D {\displaystyle D} is sometimes denoted as ◻ A B C D {\displaystyle \square ABCD} .
49-415: Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave . The interior angles of a simple (and planar ) quadrilateral ABCD add up to 360 degrees , that is This is a special case of the n -gon interior angle sum formula: S = ( n − 2) × 180° (here, n=4). All non-self-crossing quadrilaterals tile
98-401: A 2 − b 2 | . {\displaystyle K={\tfrac {1}{2}}\left|\tan \theta \right|\cdot \left|a^{2}-b^{2}\right|.} Another area formula including the sides a , b , c , d is 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
147-464: A = AB , b = BC , c = CD , d = DA , then In a convex quadrilateral ABCD with sides a = AB , b = BC , c = CD , d = DA , and where the diagonals intersect at E , where e = AE , f = BE , g = CE , and h = DE . The shape and size of a convex quadrilateral are fully determined by the lengths of its sides in sequence and of one diagonal between two specified vertices. The two diagonals p , q and
196-475: A free vector in Cartesian space equal to ( x 1 , y 1 ) and BD as ( x 2 , y 2 ) , this can be rewritten as: 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. The list applies to the most general cases, and excludes named subsets. The lengths of
245-431: A right triangle , the radius of the excircle on the hypotenuse equals the semiperimeter. The semiperimeter is the sum of the inradius and twice the circumradius. The area of the right triangle is ( s − a ) ( s − b ) {\displaystyle (s-a)(s-b)} where a, b are the legs. The formula for the semiperimeter of a quadrilateral with side lengths a, b, c, d
294-407: A convex quadrilateral are the line segments that connect opposite vertices. The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides. They intersect at the "vertex centroid" of the quadrilateral (see § Remarkable points and lines in a convex quadrilateral below). The four maltitudes of a convex quadrilateral are the perpendiculars to
343-433: A finite but nonzero area . The polygon itself is topologically equivalent to a circle , and the region outside (the exterior ) is an unbounded connected open set , with infinite area. Although the formal definition of a simple polygon is typically as a system of line segments, it is also possible (and common in informal usage) to define a simple polygon as a closed set in the plane, the union of these line segments with
392-401: A form similar to that of Heron's formula for the triangle area: Bretschneider's formula generalizes this to all convex quadrilaterals: in which α and γ are two opposite angles. The four sides of a bicentric quadrilateral are the four solutions of a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius . The area of a convex regular polygon is
441-471: A length equal to the semiperimeter. If A, B, B', C' are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency ( AA' , BB' , CC' , shown in red in the diagram) are known as splitters , and The three splitters concur at the Nagel point of the triangle. A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at
490-418: A method to explicitly construct a map from a disk to any simple polygon using specified vertex angles and pre-images of the polygon vertices on the boundary of the disk. These pre-vertices are typically computed numerically. Every finite set of points in the plane that does not lie on a single line can be connected to form the vertices of a simple polygon (allowing 180° angles); for instance, one such polygon
539-535: A parallelogram, where both pairs of opposite sides and angles are equal, this formula reduces to K = a b ⋅ sin A . {\displaystyle K=ab\cdot \sin {A}.} Alternatively, we can write the area in terms of the sides and the intersection angle θ of the diagonals, as long θ is not 90° : In the case of a parallelogram, the latter formula becomes K = 1 2 | tan θ | ⋅ |
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#1732765220695588-566: A side—through the midpoint of the opposite side. 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 . 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 . In the case of an orthodiagonal quadrilateral (e.g. rhombus, square, and kite), this formula reduces to K = p q 2 {\displaystyle K={\tfrac {pq}{2}}} since θ
637-492: A simple polygon as their result. They can be defined in a way that always produces a two-dimensional region, but this requires careful definitions of the intersection and difference operations in order to avoid creating one-dimensional features or isolated points. According to the Riemann mapping theorem , any simply connected open subset of the plane can be conformally mapped onto a disk. Schwarz–Christoffel mapping provides
686-497: A simple polygon is 2 π {\displaystyle 2\pi } . Every simple polygon with n {\displaystyle n} sides can be triangulated by n − 3 {\displaystyle n-3} of its diagonals, and by the art gallery theorem its interior is visible from some ⌊ n / 3 ⌋ {\displaystyle \lfloor n/3\rfloor } of its vertices. Simple polygons are commonly seen as
735-424: A simple polygon. The line segments that form a polygon are called its edges or sides . An endpoint of a segment is called a vertex (plural: vertices) or a corner . Edges and vertices are more formal, but may be ambiguous in contexts that also involve the edges and vertices of a graph ; the more colloquial terms sides and corners can be used to avoid this ambiguity. The number of edges always equals
784-418: A subset of its diagonals. When the polygon has n {\displaystyle n} sides, this produces n − 2 {\displaystyle n-2} triangles, separated by n − 3 {\displaystyle n-3} diagonals. The resulting partition is called a polygon triangulation . The shape of a triangulated simple polygon can be uniquely determined by
833-399: A triangle can also be calculated from the semiperimeter and side lengths: This formula can be derived from the law of sines . The inradius is The law of cotangents gives the cotangents of the half-angles at the vertices of a triangle in terms of the semiperimeter, the sides, and the inradius. The length of the internal bisector of the angle opposite the side of length a is In
882-498: A triangulation of the polygon: it is always possible to color the vertices with three colors, so that each side or diagonal in the triangulation has two endpoints of different colors. Each point of the polygon is visible to a vertex of each color, for instance one of the three vertices of the triangle containing that point in the chosen triangulation. One of the colors is used by at most ⌊ n / 3 ⌋ {\displaystyle \lfloor n/3\rfloor } of
931-492: A vertex whose two neighbors are the endpoints of a line segment that is otherwise entirely exterior to the polygon. The polygons that have exactly two ears and one mouth are called anthropomorphic polygons . According to the art gallery theorem , in a simple polygon with n {\displaystyle n} vertices, it is always possible to find a subset of at most ⌊ n / 3 ⌋ {\displaystyle \lfloor n/3\rfloor } of
980-415: Is One of the triangle area formulas involving the semiperimeter also applies to tangential quadrilaterals , which have an incircle and in which (according to Pitot's theorem ) pairs of opposite sides have lengths summing to the semiperimeter—namely, the area is the product of the inradius and the semiperimeter: The simplest form of Brahmagupta's formula for the area of a cyclic quadrilateral has
1029-459: Is 90° . The area can be also expressed in terms of bimedians as where the lengths of the bimedians are m and n and the angle between them is φ . Bretschneider's formula 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
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#17327652206951078-455: Is a closed curve in the Euclidean plane consisting of straight line segments , meeting end-to-end to form a polygonal chain . Two line segments meet at every endpoint, and there are no other points of intersection between the line segments. No proper subset of the line segments has the same properties. The qualifier simple is sometimes omitted, with the word polygon assumed to mean
1127-446: Is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter s . The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths a, b, c In any triangle, any vertex and the point where the opposite excircle touches the triangle partition the triangle's perimeter into two equal lengths, thus creating two paths each of which has
1176-500: Is positive at a convex vertex or negative at a concave vertex. For every simple polygon, the sum of the external angles is 2 π {\displaystyle 2\pi } (one full turn, 360°). Thus the sum of the internal angles, for a simple polygon with n {\displaystyle n} sides is ( n − 2 ) π {\displaystyle (n-2)\pi } . Every simple polygon can be partitioned into non-overlapping triangles by
1225-404: Is the solution to the traveling salesperson problem . Connecting points to form a polygon in this way is called polygonalization . Every simple polygon can be represented by a formula in constructive solid geometry that constructs the polygon (as a closed set including the interior) from unions and intersections of half-planes , with each side of the polygon appearing once as a half-plane in
1274-518: The Jordan curve theorem can be used to prove that such a polygon divides the plane into two regions. Indeed, Camille Jordan 's original proof of this theorem took the special case of simple polygons (stated without proof) as its starting point. The region inside the polygon (its interior ) forms a bounded set topologically equivalent to an open disk by the Jordan–Schönflies theorem , with
1323-469: 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 K = 1 2 ( a d + b c ) sin A . {\displaystyle K={\tfrac {1}{2}}(ad+bc)\sin {A}.} In
1372-549: 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 : 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 p 2 + q 2 = 2 ( m 2 + n 2 ) . {\displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).} The corresponding expressions are: if
1421-436: The diagonals in a convex quadrilateral 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. If X and Y are the feet of the normals from B and D to the diagonal AC = p in a convex quadrilateral ABCD with sides
1470-406: 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 Other, more symmetric formulas for the lengths of the diagonals, are and 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
1519-594: The form of a simple polygon. Other computational problems studied for simple polygons include constructions of the longest diagonal or the longest line segment interior to a polygon, of the convex skull (the largest convex polygon within the given simple polygon), and of various one-dimensional skeletons approximating its shape, including the medial axis and straight skeleton . Researchers have also studied producing other polygons from simple polygons using their offset curves , unions and intersections, and Minkowski sums , but these operations do not always produce
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1568-415: The formula. Converting an n {\displaystyle n} -sided polygon into this representation can be performed in time O ( n log n ) {\displaystyle O(n\log n)} . The visibility graph of a simple polygon connects its vertices by edges representing the sides and diagonals of the polygon. It always contains a Hamiltonian cycle , formed by
1617-561: The four side lengths a , b , c , d of a quadrilateral are related by the Cayley-Menger determinant , as follows: Simple polygon In geometry , a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a piecewise-linear Jordan curve consisting of finitely many line segments . These polygons include as special cases the convex polygons , star-shaped polygons , and monotone polygons . The sum of external angles of
1666-491: The input to computational geometry problems, including point in polygon testing, area computation, the convex hull of a simple polygon , triangulation, and Euclidean shortest paths . Other constructions in geometry related to simple polygons include Schwarz–Christoffel mapping , used to find conformal maps involving simple polygons, polygonalization of point sets, constructive solid geometry formulas for polygons, and visibility graphs of polygons. A simple polygon
1715-453: The interior of the polygon in a connected set. Equivalently, it is a polygon whose boundary can be partitioned into two monotone polygonal chains, subsequences of edges whose vertices, when projected perpendicularly onto L {\displaystyle L} , have the same order along L {\displaystyle L} as they do in the chain. In computational geometry , several important computational tasks involve inputs in
1764-482: The interior of the polygon. A diagonal of a simple polygon is any line segment that has two polygon vertices as its endpoints, and that otherwise is entirely interior to the polygon. The internal angle of a simple polygon, at one of its vertices, is the angle spanned by the interior of the polygon at that vertex. A vertex is convex if its internal angle is less than π {\displaystyle \pi } (a straight angle, 180°) and concave if
1813-419: The internal angle is greater than π {\displaystyle \pi } . If the internal angle is θ {\displaystyle \theta } , the external angle at the same vertex is defined to be its supplement π − θ {\displaystyle \pi -\theta } , the turning angle from one directed side to the next. The external angle
1862-405: The internal angles of the polygon and by the cross-ratios of the quadrilaterals formed by pairs of triangles that share a diagonal. According to the two ears theorem , every simple polygon that is not a triangle has at least two ears , vertices whose two neighbors are the endpoints of a diagonal. A related theorem states that every simple polygon that is not a convex polygon has a mouth ,
1911-475: The lengths of two bimedians and one diagonal are given, and if the lengths of two diagonals and one bimedian are given. 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
1960-465: The midpoint of one of the three sides. So any cleaver, like any splitter, divides the triangle into two paths each of whose length equals the semiperimeter. The three cleavers concur at the center of the Spieker circle , which is the incircle of the medial triangle ; the Spieker center is the center of mass of all the points on the triangle's edges. A line through the triangle's incenter bisects
2009-419: The number of vertices. Some sources allow two line segments to form a straight angle (180°), while others disallow this, instead requiring collinear segments of a closed polygonal chain to be merged into a single longer side. Two vertices are neighbors if they are the two endpoints of one of the sides of the polygon. Simple polygons are sometimes called Jordan polygons , because they are Jordan curves ;
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2058-507: The perimeter if and only if it also bisects the area. A triangle's semiperimeter equals the perimeter of its medial triangle . By the triangle inequality , the longest side length of a triangle is less than the semiperimeter. The area A of any triangle is the product of its inradius (the radius of its inscribed circle) and its semiperimeter: The area of a triangle can also be calculated from its semiperimeter and side lengths a, b, c using Heron's formula : The circumradius R of
2107-402: The plane , by repeated rotation around the midpoints of their edges. Any quadrilateral that is not self-intersecting is a simple quadrilateral. In a convex quadrilateral all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral. [REDACTED] In a concave quadrilateral, one interior angle is bigger than 180°, and one of the two diagonals lies outside
2156-481: The polygon sides. The computational complexity of reconstructing a polygon that has a given graph as its visibility graph, with a specified Hamiltonian cycle as its cycle of sides, remains an open problem. Semiperimeter#Quadrilaterals In geometry , the semiperimeter of a polygon is half its perimeter . Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it
2205-407: The quadrilateral. A self-intersecting quadrilateral is called variously a cross-quadrilateral , crossed quadrilateral , butterfly quadrilateral or bow-tie quadrilateral . 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°. The two diagonals of
2254-452: The sides a , b , c , d and the angle α (between a and b ) is: which can also be used for the area of a concave quadrilateral (having the concave part opposite to angle α ), by just changing the first sign + to - . The following two formulas express the area in terms of the sides a , b , c and d , the semiperimeter s , and the diagonals p , q : The first reduces to Brahmagupta's formula in
2303-405: The square of the line segment connecting the midpoints of the diagonals. Thus where x is the distance between the midpoints of the diagonals. 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
2352-415: The vertices with the property that every point in the polygon is visible from one of the selected vertices. This means that, for each point p {\displaystyle p} in the polygon, there exists a line segment connecting p {\displaystyle p} to a selected vertex, passing only through interior points of the polygon. One way to prove this is to use graph coloring on
2401-466: The vertices, proving the theorem. Every convex polygon is a simple polygon. Another important class of simple polygons are the star-shaped polygons , the polygons that have a point (interior or on their boundary) from which every point is visible. A monotone polygon , with respect to a straight line L {\displaystyle L} , is a polygon for which every line perpendicular to L {\displaystyle L} intersects
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