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108-420: In geometry , calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is T = b h / 2 , {\displaystyle T=bh/2,} where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes

216-496: A {\displaystyle h_{a}} ⁠ , ⁠ h b {\displaystyle h_{b}} ⁠ , and ⁠ h c {\displaystyle h_{c}} ⁠ are the altitudes from sides ⁠ a , {\displaystyle a,} ⁠ ⁠ b , {\displaystyle b,} ⁠ and ⁠ c {\displaystyle c} ⁠ respectively, and semi-sum of their reciprocals

324-486: A {\displaystyle \mathbf {b} -\mathbf {a} } ⁠ and ⁠ c − a {\displaystyle \mathbf {c} -\mathbf {a} } ⁠ are the translation vectors from vertex ⁠ A {\displaystyle A} ⁠ to each of the others, and ⁠ ∧ {\displaystyle \wedge } ⁠ is the wedge product . If vertex ⁠ A {\displaystyle A} ⁠

432-478: A 2 = h 2 + ( c − d ) 2 {\displaystyle a^{2}=h^{2}+(c-d)^{2}} according to the figure at the right. Subtracting these yields a 2 − b 2 = c 2 − 2 c d . {\displaystyle a^{2}-b^{2}=c^{2}-2cd.} This equation allows us to express ⁠ d {\displaystyle d} ⁠ in terms of

540-463: A ¯ {\displaystyle {\bar {a}}} , b ¯ {\displaystyle {\bar {b}}} , and c ¯ {\displaystyle {\bar {c}}} , then the formula is equivalent to the shoelace formula. In three dimensions, the area of a general triangle A = ( x A , y A , z A ) , B = ( x B , y B , z B ) and C = ( x C , y C , z C )

648-417: A + b + c ) = {\displaystyle s={\tfrac {1}{2}}(a+b+c)={}} 1 2 ( 4 + 13 + 15 ) = 16 {\displaystyle {\tfrac {1}{2}}(4+13+15)=16} and so the area is In this example, the side lengths and area are integers , making it a Heronian triangle . However, Heron's formula works equally well in cases where one or more of

756-460: A , {\displaystyle a,} ⁠ ⁠ b , {\displaystyle b,} ⁠ and ⁠ c {\displaystyle c} ⁠ respectively, and their semi-sum is σ = 1 2 ( m a + m b + m c ) , {\displaystyle \sigma ={\tfrac {1}{2}}(m_{a}+m_{b}+m_{c}),} then Next, if ⁠ h

864-408: A , {\displaystyle a,} ⁠ ⁠ b , {\displaystyle b,} ⁠ and ⁠ c {\displaystyle c} ⁠ the semiperimeter s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} and area ⁠ S {\displaystyle S} ⁠ , such

972-461: A cyclic quadrilateral whose sides have lengths ⁠ a , {\displaystyle a,} ⁠ ⁠ b , {\displaystyle b,} ⁠ ⁠ c , {\displaystyle c,} ⁠ ⁠ d {\displaystyle d} ⁠ as where s = 1 2 ( a + b + c + d ) {\displaystyle s={\tfrac {1}{2}}(a+b+c+d)}

1080-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as

1188-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to

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1296-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes

1404-446: A collection of over a hundred distinct area formulas for the triangle. These include: for circumradius (radius of the circumcircle) R , and Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure') is a branch of mathematics concerned with properties of space such as

1512-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,

1620-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In

1728-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,

1836-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry

1944-451: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in

2052-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received

2160-528: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to

2268-457: A polygon is given directly by r i r i +1 sin(θ i +1 − θ i )/2 . This is valid for all values of θ, with some decrease in numerical accuracy when |θ| is many orders of magnitude greater than π. With this formulation negative area indicates clockwise traversal, which should be kept in mind when mixing polar and cartesian coordinates. Just as the choice of y -axis ( x = 0 ) is immaterial for line integration in cartesian coordinates, so

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2376-602: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies

2484-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating

2592-403: A special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral). A modern proof, which uses algebra and is quite different from the one provided by Heron, follows. Let ⁠ a , {\displaystyle a,} ⁠ ⁠ b , {\displaystyle b,} ⁠ ⁠ c {\displaystyle c} ⁠ be

2700-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c.  1900 , with

2808-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in

2916-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of

3024-448: A vector ⁠ b {\displaystyle \mathbf {b} } ⁠ with coordinates ⁠ ( x B , y B ) {\displaystyle (x_{B},y_{B})} ⁠ and vector ⁠ c {\displaystyle \mathbf {c} } ⁠ with coordinates ⁠ ( x C , y C ) {\displaystyle (x_{C},y_{C})} ⁠ ,

3132-570: Is H = 1 2 ( h a − 1 + h b − 1 + h c − 1 ) , {\displaystyle H={\tfrac {1}{2}}{\bigl (}h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1}{\bigr )},} then Finally, if ⁠ α , {\displaystyle \alpha ,} ⁠ ⁠ β , {\displaystyle \beta ,} ⁠ and ⁠ γ {\displaystyle \gamma } ⁠ are

3240-554: Is cot ⁡ α 2 + cot ⁡ β 2 + cot ⁡ γ 2 = {\textstyle \cot {\tfrac {\alpha }{2}}+\cot {\tfrac {\beta }{2}}+\cot {\tfrac {\gamma }{2}}={}} cot ⁡ α 2 cot ⁡ β 2 cot ⁡ γ 2 , {\displaystyle \cot {\tfrac {\alpha }{2}}\cot {\tfrac {\beta }{2}}\cot {\tfrac {\gamma }{2}},}

3348-657: Is where s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} is the semiperimeter. The triangle can alternately be broken into six triangles (in congruent pairs) of altitude ⁠ r {\displaystyle r} ⁠ and bases ⁠ s − a , {\displaystyle s-a,} ⁠ ⁠ s − b , {\displaystyle s-b,} ⁠ and ⁠ s − c {\displaystyle s-c} ⁠ of combined area (see law of cotangents ) The middle step above

Area of a triangle - Misplaced Pages Continue

3456-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing

3564-426: Is numerically unstable for triangles with a very small angle when using floating-point arithmetic . A stable alternative involves arranging the lengths of the sides so that a ≥ b ≥ c {\displaystyle a\geq b\geq c} and computing The extra brackets indicate the order of operations required to achieve numerical stability in the evaluation. Three other formulae for

3672-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in

3780-463: Is a special case of Brahmagupta's formula for the area of a cyclic quadrilateral . Heron's formula and Brahmagupta's formula are both special cases of Bretschneider's formula for the area of a quadrilateral . Heron's formula can be obtained from Brahmagupta's formula or Bretschneider's formula by setting one of the sides of the quadrilateral to zero. Brahmagupta's formula gives the area ⁠ K {\displaystyle K} ⁠ of

3888-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood

3996-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are

4104-405: Is half of the area of a parallelogram : where ⁠ a {\displaystyle \mathbf {a} } ⁠ , ⁠ b {\displaystyle \mathbf {b} } ⁠ , and ⁠ c {\displaystyle \mathbf {c} } ⁠ are vectors to the triangle's vertices from any arbitrary origin point, so that ⁠ b −

4212-460: Is known as Heron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book, Metrica , written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician Euclid proved that

4320-510: Is not always the case. For example, the land surveyor of a triangular field might find it relatively easy to measure the length of each side, but relatively difficult to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. Other frequently used formulas for the area of a triangle use trigonometry , side lengths ( Heron's formula ), vectors, coordinates , line integrals , Pick's theorem , or other properties. Heron of Alexandria found what

4428-437: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given

Area of a triangle - Misplaced Pages Continue

4536-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study

4644-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,

4752-648: Is taken to be the origin, this simplifies to 1 2 ‖ b ∧ c ‖ {\displaystyle {\tfrac {1}{2}}\|\mathbf {b} \wedge \mathbf {c} \|} . The oriented relative area of a parallelogram in any affine space, a type of bivector , is defined as ⁠ u ∧ v {\displaystyle \mathbf {u} \wedge \mathbf {v} } ⁠ where ⁠ u {\displaystyle \mathbf {u} } ⁠ and ⁠ v {\displaystyle \mathbf {v} } ⁠ are translation vectors from one vertex of

4860-496: Is the Pythagorean sum of the areas of the respective projections on the three principal planes (i.e. x = 0, y = 0 and z = 0): The area within any closed curve, such as a triangle, is given by the line integral around the curve of the algebraic or signed distance of a point on the curve from an arbitrary oriented straight line L . Points to the right of L as oriented are taken to be at negative distance from L , while

4968-425: Is the semiperimeter , or half of the triangle's perimeter. Three other equivalent ways of writing Heron's formula are Three formulas have the same structure as Heron's formula but are expressed in terms of different variables. First, denoting the medians from sides a , b , and c respectively as m a , m b , and m c and their semi-sum ( m a + m b + m c )/2 as σ, we have Next, denoting

5076-420: Is the semiperimeter . Heron's formula is also a special case of the formula for the area of a trapezoid or trapezium based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero. Expressing Heron's formula with a Cayley–Menger determinant in terms of the squares of the distances between the three given vertices, illustrates its similarity to Tartaglia's formula for

5184-413: Is the choice of zero heading ( θ = 0 ) immaterial here. See Pick's theorem for a technique for finding the area of any arbitrary lattice polygon (one drawn on a grid with vertically and horizontally adjacent lattice points at equal distances, and with vertices on lattice points). The theorem states: where I {\displaystyle I} is the number of internal lattice points and B

5292-427: Is the diameter of the circumcircle , D = a / sin ⁡ α = b / sin ⁡ β = c / sin ⁡ γ . {\displaystyle D=a/{\sin \alpha }=b/{\sin \beta }=c/{\sin \gamma }.} This last formula coincides with the standard Heron formula when the circumcircle has unit diameter. Heron's formula

5400-583: Is the number of lattice points lying on the border of the polygon. Numerous other area formulas exist, such as where r is the inradius , and s is the semiperimeter (in fact, this formula holds for all tangential polygons ), and where r a , r b , r c {\displaystyle r_{a},\,r_{b},\,r_{c}} are the radii of the excircles tangent to sides a, b, c respectively. We also have and for circumdiameter D ; and for angle α ≠ 90°. The area can also be expressed as In 1885, Baker gave

5508-753: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs

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5616-466: The Cayley–Menger determinant , The formula is credited to Heron (or Hero) of Alexandria ( fl. 60 AD), and a proof can be found in his book Metrica . Mathematical historian Thomas Heath suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates

5724-690: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c.  1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated

5832-523: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c.  1230  – c.  1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first

5940-518: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with

6048-509: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of

6156-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,

6264-435: The triple cotangent identity , which applies because the sum of half-angles is α 2 + β 2 + γ 2 = π 2 . {\textstyle {\tfrac {\alpha }{2}}+{\tfrac {\beta }{2}}+{\tfrac {\gamma }{2}}={\tfrac {\pi }{2}}.} Combining the two, we get from which the result follows. Heron's formula as given above

6372-596: The volume of a three-simplex . Another generalization of Heron's formula to pentagons and hexagons inscribed in a circle was discovered by David P. Robbins . If ⁠ U , {\displaystyle U,} ⁠ ⁠ V , {\displaystyle V,} ⁠ ⁠ W , {\displaystyle W,} ⁠ ⁠ u , {\displaystyle u,} ⁠ ⁠ v , {\displaystyle v,} ⁠ ⁠ w {\displaystyle w} ⁠ are lengths of edges of

6480-631: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing

6588-496: The 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into

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6696-474: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by

6804-736: The Greeks. It was published in 1247 in Shushu Jiuzhang (" Mathematical Treatise in Nine Sections "), written by Qin Jiushao . The height of a triangle can be found through the application of trigonometry . Using the labels in the image on the right, the altitude is h = a sin γ {\displaystyle \gamma } . Substituting this in the formula T = 1 2 b h {\displaystyle T={\tfrac {1}{2}}bh} derived above,

6912-422: The altitudes from sides a , b , and c respectively as h a , h b , and h c , and denoting the semi-sum of the reciprocals of the altitudes as H = ( h a − 1 + h b − 1 + h c − 1 ) / 2 {\displaystyle H=(h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1})/2} we have And denoting

7020-531: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in

7128-446: The area is an overall indicator of the direction of traversal, with negative area indicating counterclockwise traversal. The area of a triangle then falls out as the case of a polygon with three sides. While the line integral method has in common with other coordinate-based methods the arbitrary choice of a coordinate system, unlike the others it makes no arbitrary choice of vertex of the triangle as origin or of side as base. Furthermore,

7236-421: The area of a general triangle have a similar structure as Heron's formula, expressed in terms of different variables. First, if ⁠ m a , {\displaystyle m_{a},} ⁠ ⁠ m b , {\displaystyle m_{b},} ⁠ and ⁠ m c {\displaystyle m_{c}} ⁠ are the medians from sides ⁠

7344-526: The area of a triangle from its height: If ⁠ r {\displaystyle r} ⁠ is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of equal altitude ⁠ r {\displaystyle r} ⁠ and bases ⁠ a , {\displaystyle a,} ⁠ ⁠ b , {\displaystyle b,} ⁠ and ⁠ c . {\displaystyle c.} ⁠ Their combined area

7452-539: The area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry . In 499 Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , expressed the area of a triangle as one-half the base times the height in the Aryabhatiya . A formula equivalent to Heron's was discovered by the Chinese independently of

7560-411: The area of the triangle can be expressed as: (where α is the interior angle at A , β is the interior angle at B , γ {\displaystyle \gamma } is the interior angle at C and c is the line AB ). Furthermore, since sin α = sin ( π − α) = sin (β + γ {\displaystyle \gamma } ), and similarly for the other two angles: and analogously if

7668-491: The choice of coordinate system defined by L commits to only two degrees of freedom rather than the usual three, since the weight is a local distance (e.g. x i +1 − x i in the above) whence the method does not require choosing an axis normal to L . When working in polar coordinates it is not necessary to convert to Cartesian coordinates to use line integration, since the line integral between consecutive vertices ( r i ,θ i ) and ( r i +1 ,θ i +1 ) of

7776-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry

7884-513: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c.  287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of

7992-468: The distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model

8100-520: The equation is: which can be written as If the points are labeled sequentially in the counterclockwise direction, the above determinant expressions are positive and the absolute value signs can be omitted. The above formula is known as the shoelace formula or the surveyor's formula. If we locate the vertices in the complex plane and denote them in counterclockwise sequence as a = x A + y A i , b = x B + y B i , and c = x C + y C i , and denote their complex conjugates as

8208-428: The field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits

8316-520: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c.  355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as

8424-526: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between

8532-598: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning

8640-537: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including

8748-391: The known side is a or c . and analogously if the known side is b or c . A triangle's shape is uniquely determined by the lengths of the sides, so its metrical properties, including area, can be described in terms of those lengths. By Heron's formula , where s = 1 2 ( a + b + c ) {\textstyle s={\tfrac {1}{2}}(a+b+c)}

8856-552: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived

8964-418: The length of a perpendicular from the vertex opposite the base onto the line containing the base. Euclid proved that the area of a triangle is half that of a parallelogram with the same base and height in his book Elements in 300 BCE. In 499 CE Aryabhata , used this illustrated method in the Aryabhatiya (section 2.6). Although simple, this formula is only useful if the height can be readily found, which

9072-405: The magnitude of the wedge product is (See the following section.) If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = ( x B , y B ) and C = ( x C , y C ) , then the area can be computed as 1 ⁄ 2 times the absolute value of the determinant For three general vertices,

9180-411: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of

9288-429: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,

9396-442: The nature of geometric structures modelled on, or arising out of, the complex plane . Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables , and has found applications to string theory and mirror symmetry . Heron%27s formula It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work Metrica , though it

9504-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of

9612-407: The parallelogram to each of the two adacent vertices. In Euclidean space, the magnitude of this bivector is a well-defined scalar number representing the area of the parallelogram. (For vectors in three-dimensional space, the bivector-valued wedge product has the same magnitude as the vector-valued cross product , but unlike the cross product, which is only defined in three-dimensional Euclidean space,

9720-514: The physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem , a problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During

9828-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of

9936-424: The previous one using the elementary vector identity u 2 v 2 = ( u ⋅ v ) 2 + ‖ u ∧ v ‖ 2 {\displaystyle \mathbf {u} ^{2}\mathbf {v} ^{2}=(\mathbf {u} \cdot \mathbf {v} )^{2}+\|\mathbf {u} \wedge \mathbf {v} \|^{2}} . In two-dimensional Euclidean space, for

10044-482: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that

10152-549: The reference given in that work. A formula equivalent to Heron's was discovered by the Chinese: published in Mathematical Treatise in Nine Sections ( Qin Jiushao , 1247). There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, or as a special case of De Gua's theorem (for the particular case of acute triangles), or as

10260-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid

10368-472: The semi-sum of the angles' sines as S = [(sin α) + (sin β) + (sin γ)]/2 , we have where D is the diameter of the circumcircle : D = a sin ⁡ α = b sin ⁡ β = c sin ⁡ γ . {\displaystyle D={\tfrac {a}{\sin \alpha }}={\tfrac {b}{\sin \beta }}={\tfrac {c}{\sin \gamma }}.} The area of triangle ABC

10476-554: The side lengths are not integers. Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways, After expansion, the expression under the square root is a quadratic polynomial of the squared side lengths ⁠ a 2 {\displaystyle a^{2}} ⁠ , ⁠ b 2 {\displaystyle b^{2}} ⁠ , ⁠ c 2 {\displaystyle c^{2}} ⁠ . The same relation can be expressed using

10584-415: The sides of the triangle and ⁠ α , {\displaystyle \alpha ,} ⁠ ⁠ β , {\displaystyle \beta ,} ⁠ ⁠ γ {\displaystyle \gamma } ⁠ the angles opposite those sides. Applying the law of cosines we get From this proof, we get the algebraic statement that The altitude of

10692-426: The sides of the triangle: For the height of the triangle we have that h 2 = b 2 − d 2 . {\displaystyle h^{2}=b^{2}-d^{2}.} By replacing ⁠ d {\displaystyle d} ⁠ with the formula given above and applying the difference of squares identity we get We now apply this result to the formula that calculates

10800-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry

10908-404: The tetrahedron (first three form a triangle; ⁠ u {\displaystyle u} ⁠ opposite to ⁠ U {\displaystyle U} ⁠ and so on), then where There are also formulas for the area of a triangle in terms of its side lengths for triangles in the sphere or the hyperbolic plane . For a triangle in the sphere with side lengths ⁠

11016-409: The theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and

11124-407: The three angle measures of the triangle, and the semi-sum of their sines is S = 1 2 ( sin ⁡ α + sin ⁡ β + sin ⁡ γ ) , {\displaystyle S={\tfrac {1}{2}}(\sin \alpha +\sin \beta +\sin \gamma ),} then where ⁠ D {\displaystyle D} ⁠

11232-444: The triangle on base ⁠ a {\displaystyle a} ⁠ has length ⁠ b sin ⁡ γ {\displaystyle b\sin \gamma } ⁠ , and it follows The following proof is very similar to one given by Raifaizen. By the Pythagorean theorem we have b 2 = h 2 + d 2 {\displaystyle b^{2}=h^{2}+d^{2}} and

11340-748: The wedge product is well-defined in an affine space of any dimension.) The area of triangle ABC can also be expressed in terms of dot products . Taking vertex ⁠ A {\displaystyle A} ⁠ to be the origin and calling translation vectors to the other vertices ⁠ b {\displaystyle \mathbf {b} } ⁠ and ⁠ c {\displaystyle \mathbf {c} } ⁠ , where for any Euclidean vector v 2 = ‖ v ‖ 2 = v ⋅ v {\displaystyle \mathbf {v} ^{2}=\|\mathbf {v} \|^{2}=\mathbf {v} \cdot \mathbf {v} } . This area formula can be derived from

11448-449: The weight for the integral is taken to be the component of arc length parallel to L rather than arc length itself. This method is well suited to computation of the area of an arbitrary polygon . Taking L to be the x -axis, the line integral between consecutive vertices ( x i , y i ) and ( x i +1 , y i +1 ) is given by the base times the mean height, namely ( x i +1 − x i )( y i + y i +1 )/2 . The sign of

11556-441: Was probably known centuries earlier. Let ⁠ △ A B C {\displaystyle \triangle ABC} ⁠ be the triangle with sides a = 4 , {\displaystyle a=4,} b = 13 , {\displaystyle b=13,} and c = 15. {\displaystyle c=15.} This triangle's semiperimeter is s = 1 2 (

11664-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in

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