Areais themeasureof aregion's size on asurface.The area of a plane region orplane arearefers to the area of ashapeorplanar lamina,whilesurface arearefers to the area of anopen surfaceor theboundaryof athree-dimensional object.Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount ofpaintnecessary to cover the surface with a single coat.[1]It is the two-dimensional analogue of thelengthof acurve(a one-dimensional concept) or thevolumeof a solid (a three-dimensional concept). Two different regions may have the same area (as insquaring the circle); bysynecdoche,"area" sometimes is used to refer to the region, as in a "polygonal area".
Area | |
---|---|
![]() The areas of this square and thisdiskare the same. | |
Common symbols | AorS |
SI unit | Square metre[m2] |
InSI base units | 1m2 |
Dimension |
The area of a shape can be measured by comparing the shape tosquaresof a fixed size.[2]In theInternational System of Units(SI), the standard unit of area is thesquare metre(written as m2), which is the area of a square whose sides are onemetrelong.[3]A shape with an area of three square metres would have the same area as three such squares. Inmathematics,theunit squareis defined to have area one, and the area of any other shape or surface is adimensionlessreal number.
There are several well-knownformulasfor the areas of simple shapes such astriangles,rectangles,andcircles.Using these formulas, the area of anypolygoncan be found bydividing the polygon into triangles.[4]For shapes with curved boundary,calculusis usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for thehistorical development of calculus.[5]
For a solid shape such as asphere,cone, or cylinder, the area of its boundary surface is called thesurface area.[1][6][7]Formulas for the surface areas of simple shapes were computed by theancient Greeks,but computing the surface area of a more complicated shape usually requiresmultivariable calculus.
Area plays an important role in modern mathematics. In addition to its obvious importance ingeometryand calculus, area is related to the definition ofdeterminantsinlinear algebra,and is a basic property of surfaces indifferential geometry.[8]Inanalysis,the area of a subset of the plane is defined usingLebesgue measure,[9]though not every subset is measurable if one supposes the axiom of choice.[10]In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.[1]
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.
Formal definition
editAn approach to defining what is meant by "area" is throughaxioms."Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties:[11]
- For allSinM,a(S) ≥ 0.
- IfSandTare inMthen so areS∪TandS∩T,and alsoa(S∪T) =a(S) +a(T) −a(S∩T).
- IfSandTare inMwithS⊆TthenT−Sis inManda(T−S) =a(T) −a(S).
- If a setSis inMandSis congruent toTthenTis also inManda(S) =a(T).
- Every rectangleRis inM.If the rectangle has lengthhand breadthkthena(R) =hk.
- LetQbe a set enclosed between two step regionsSandT.A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e.S⊆Q⊆T.If there is a unique numbercsuch thata(S) ≤ c ≤a(T)for all such step regionsSandT,thena(Q) =c.
It can be proved that such an area function actually exists.[12]
Units
editEveryunit of lengthhas a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured insquare metres(m2), square centimetres (cm2), square millimetres (mm2),square kilometres(km2),square feet(ft2),square yards(yd2),square miles(mi2), and so forth.[13]Algebraically, these units can be thought of as thesquaresof the corresponding length units.
The SI unit of area is the square metre, which is considered anSI derived unit.[3]
Conversions
editCalculation of the area of a square whose length and width are 1 metre would be:
1 metre × 1 metre = 1 m2
and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as:
3 metres × 2 metres = 6 m2.This is equivalent to 6 million square millimetres. Other useful conversions are:
- 1 square kilometre =1,000,000square metres
- 1 square metre =10,000square centimetres = 1,000,000 square millimetres
- 1 square centimetre =100square millimetres.
Non-metric units
editIn non-metric units, the conversion between two square units is thesquareof the conversion between the corresponding length units.
the relationship between square feet and square inches is
- 1 square foot = 144 square inches,
where 144 = 122= 12 × 12. Similarly:
- 1 square yard =9square feet
- 1 square mile = 3,097,600 square yards = 27,878,400 square feet
In addition, conversion factors include:
- 1 square inch = 6.4516 square centimetres
- 1 square foot =0.09290304square metres
- 1 square yard =0.83612736square metres
- 1 square mile =2.589988110336square kilometres
Other units including historical
editThere are several other common units for area. Thearewas the original unit of area in themetric system,with:
- 1 are = 100 square metres
Though the are has fallen out of use, thehectareis still commonly used to measure land:[13]
- 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometres
Other uncommon metric units of area include thetetrad,thehectad,and themyriad.
Theacreis also commonly used to measure land areas, where
- 1 acre = 4,840 square yards = 43,560 square feet.
An acre is approximately 40% of a hectare.
On the atomic scale, area is measured in units ofbarns,such that:[13]
- 1 barn = 10−28square meters.
The barn is commonly used in describing the cross-sectional area of interaction innuclear physics.[13]
InSouth Asia(mainly Indians), although the countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used.[14][15][16][17]
Some traditional South Asian units that have fixed value:
- 1 Killa = 1 acre
- 1 Ghumaon = 1 acre
- 1 Kanal = 0.125 acre (1 acre = 8 kanal)
- 1 Decimal = 48.4 square yards
- 1 Chatak = 180 square feet
History
editCircle area
editIn the 5th century BCE,Hippocrates of Chioswas the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of hisquadratureof thelune of Hippocrates,[18]but did not identify theconstant of proportionality.Eudoxus of Cnidus,also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared.[19]
Subsequently, Book I ofEuclid'sElementsdealt with equality of areas between two-dimensional figures. The mathematicianArchimedesused the tools ofEuclidean geometryto show that the area inside a circle is equal to that of aright trianglewhose base has the length of the circle's circumference and whose height equals the circle's radius, in his bookMeasurement of a Circle.(The circumference is 2πr,and the area of a triangle is half the base times the height, yielding the areaπr2for the disk.) Archimedes approximated the value of π (and hence the area of a unit-radius circle) withhis doubling method,in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regularhexagon,then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same withcircumscribed polygons).
Triangle area
editHeron of Alexandriafound what is known asHeron's formulafor 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 thatArchimedesknew the formula over two centuries earlier,[20]and sinceMetricais a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.[21]In 300 BCE Greek mathematicianEuclidproved that the area of a triangle is half that of a parallelogram with the same base and height in his bookElements of Geometry.[22]
In 499Aryabhata,a greatmathematician-astronomerfrom the classical age ofIndian mathematicsandIndian astronomy,expressed the area of a triangle as one-half the base times the height in theAryabhatiya.[23]
A formula equivalent to Heron's was discovered by the Chinese independently of the Greeks. It was published in 1247 inShushu Jiuzhang( "Mathematical Treatise in Nine Sections"), written byQin Jiushao.[24]Quadrilateral area
editIn the 7th century CE,Brahmaguptadeveloped a formula, now known asBrahmagupta's formula,for the area of acyclic quadrilateral(aquadrilateralinscribedin a circle) in terms of its sides. In 1842, the German mathematiciansCarl Anton BretschneiderandKarl Georg Christian von Staudtindependently found a formula, known asBretschneider's formula,for the area of any quadrilateral.
General polygon area
editThe development ofCartesian coordinatesbyRené Descartesin the 17th century allowed the development of thesurveyor's formulafor the area of any polygon with knownvertexlocations byGaussin the 19th century.
Areas determined using calculus
editThe development ofintegral calculusin the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of anellipseand thesurface areasof various curved three-dimensional objects.
Area formulas
editPolygon formulas
editFor a non-self-intersecting (simple) polygon, theCartesian coordinates(i=0, 1,...,n-1) of whosenverticesare known, the area is given by thesurveyor's formula:[25]
where wheni=n-1, theni+1 is expressed asmodulusnand so refers to 0.
Rectangles
editThe most basic area formula is the formula for the area of arectangle.Given a rectangle with lengthland widthw,the formula for the area is:[2]
- A=lw(rectangle).
That is, the area of the rectangle is the length multiplied by the width. As a special case, asl=win the case of a square, the area of a square with side lengthsis given by the formula:[1][2]
- A=s2(square).
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as adefinitionoraxiom.On the other hand, ifgeometryis developed beforearithmetic,this formula can be used to definemultiplicationofreal numbers.
Dissection, parallelograms, and triangles
editMost other simple formulas for area follow from the method ofdissection. This involves cutting a shape into pieces, whose areas mustsumto the area of the original shape. For an example, anyparallelogramcan be subdivided into atrapezoidand aright triangle,as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:[2]
- A=bh(parallelogram).
However, the same parallelogram can also be cut along adiagonalinto twocongruenttriangles, as shown in the figure to the right. It follows that the area of eachtriangleis half the area of the parallelogram:[2]
- (triangle).
Similar arguments can be used to find area formulas for thetrapezoid[26]as well as more complicatedpolygons.[27]
Area of curved shapes
editCircles
editThe formula for the area of acircle(more properly called the area enclosed by a circle or the area of adisk) is based on a similar method. Given a circle of radiusr,it is possible to partition the circle intosectors,as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram isr,and the width is half thecircumferenceof the circle, orπr.Thus, the total area of the circle isπr2:[2]
- A= πr2(circle).
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. Thelimitof the areas of the approximate parallelograms is exactlyπr2,which is the area of the circle.[28]
This argument is actually a simple application of the ideas ofcalculus.In ancient times, themethod of exhaustionwas used in a similar way to find the area of the circle, and this method is now recognized as a precursor tointegral calculus.Using modern methods, the area of a circle can be computed using adefinite integral:
Ellipses
editThe formula for the area enclosed by anellipseis related to the formula of a circle; for an ellipse withsemi-majorandsemi-minoraxesxandythe formula is:[2]
Non-planar surface area
editMost basic formulas forsurface areacan be obtained by cutting surfaces and flattening them out (see:developable surfaces). For example, if the side surface of acylinder(or anyprism) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of acone,the side surface can be flattened out into a sector of a circle, and the resulting area computed.
The formula for the surface area of asphereis more difficult to derive: because a sphere has nonzeroGaussian curvature,it cannot be flattened out. The formula for the surface area of a sphere was first obtained byArchimedesin his workOn the Sphere and Cylinder.The formula is:[6]
- A= 4πr2(sphere),
whereris the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar tocalculus.
General formulas
editAreas of 2-dimensional figures
edit- Atriangle:(whereBis any side, andhis the distance from the line on whichBlies to the other vertex of the triangle). This formula can be used if the heighthis known. If the lengths of the three sides are known thenHeron's formulacan be used:wherea,b,care the sides of the triangle, andis half of its perimeter.[2]If an angle and its two included sides are given, the area iswhereCis the given angle andaandbare its included sides.[2]If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of.This formula is also known as theshoelace formulaand is an easy way to solve for the area of a coordinate triangle by substituting the 3 points(x1,y1),(x2,y2),and(x3,y3).The shoelace formula can also be used to find the areas of other polygons when their vertices are known. Another approach for a coordinate triangle is to usecalculusto find the area.
- Asimple polygonconstructed on a grid of equal-distanced points (i.e., points withintegercoordinates) such that all the polygon's vertices are grid points:,whereiis the number of grid points inside the polygon andbis the number of boundary points. This result is known asPick's theorem.[29]
Area in calculus
edit- The area between a positive-valued curve and the horizontal axis, measured between two valuesaandb(b is defined as the larger of the two values) on the horizontal axis, is given by the integral fromatobof the function that represents the curve:[1]
- The area between thegraphsof two functions isequalto theintegralof onefunction,f(x),minusthe integral of the other function,g(x):
- whereis the curve with the greater y-value.
- An area bounded by a functionexpressed inpolar coordinatesis:[1]
- The area enclosed by aparametric curvewith endpointsis given by theline integrals:
- or thez-component of
- (For details, seeGreen's theorem § Area calculation.) This is the principle of theplanimetermechanical device.
Bounded area between two quadratic functions
editTo find the bounded area between twoquadratic functions,we first subtract one from the other, writing the difference as wheref(x) is the quadratic upper bound andg(x) is the quadratic lower bound. By the area integral formulas above andVieta's formula,we can obtain that[30][31] The above remains valid if one of the bounding functions is linear instead of quadratic.
Surface area of 3-dimensional figures
edit- Cone:[32],whereris the radius of the circular base, andhis the height. That can also be rewritten as[32]orwhereris the radius andlis the slant height of the cone.is the base area whileis the lateral surface area of the cone.[32]
- Cube:,wheresis the length of an edge.[6]
- Cylinder:,whereris the radius of a base andhis the height. Thecan also be rewritten as,wheredis the diameter.
- Prism:,whereBis the area of a base,Pis the perimeter of a base, andhis the height of the prism.
- pyramid:,whereBis the area of the base,Pis the perimeter of the base, andLis the length of the slant.
- Rectangular prism:,whereis the length,wis the width, andhis the height.
General formula for surface area
editThe general formula for the surface area of the graph of a continuously differentiable functionwhereandis a region in the xy-plane with the smooth boundary:
An even more general formula for the area of the graph of aparametric surfacein the vector formwhereis a continuously differentiable vector function ofis:[8]
List of formulas
editShape | Formula | Variables |
---|---|---|
Square | ||
Rectangle | ||
Triangle | ||
Triangle | ||
Triangle |
||
Isosceles triangle | ||
Regulartriangle |
||
Rhombus/Kite | ||
Parallelogram | ||
Trapezoid | ||
Regularhexagon | ||
Regularoctagon | ||
Regular polygon (sides) |
|
(perimeter) |
Circle | (diameter) |
|
Circular sector | ||
Ellipse | ||
Integral | ||
Surface area | ||
Sphere |
||
Cuboid | ||
Cylinder (incl. bottom and top) |
||
Cone (incl. bottom) |
||
Torus | ||
Surface of revolution | (rotation around the x-axis) |
The above calculations show how to find the areas of many commonshapes.
The areas of irregular (and thus arbitrary) polygons can be calculated using the "Surveyor's formula"(shoelace formula).[28]
Relation of area to perimeter
editTheisoperimetric inequalitystates that, for a closed curve of lengthL(so the region it encloses hasperimeterL) and for areaAof the region that it encloses,
and equality holds if and only if the curve is acircle.Thus a circle has the largest area of any closed figure with a given perimeter.
At the other extreme, a figure with given perimeterLcould have an arbitrarily small area, as illustrated by arhombusthat is "tipped over" arbitrarily far so that two of itsanglesare arbitrarily close to 0° and the other two are arbitrarily close to 180°.
For a circle, the ratio of the area to thecircumference(the term for the perimeter of a circle) equals half theradiusr.This can be seen from the area formulaπr2and the circumference formula 2πr.
The area of aregular polygonis half its perimeter times theapothem(where the apothem is the distance from the center to the nearest point on any side).
Fractals
editDoubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). But if the one-dimensional lengths of afractaldrawn in two dimensions are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This power is called thefractal dimensionof the fractal. [33]
Area bisectors
editThere are an infinitude of lines that bisect the area of a triangle. Three of them are themediansof the triangle (which connect the sides' midpoints with the opposite vertices), and these areconcurrentat the triangle'scentroid;indeed, they are the only area bisectors that go through the centroid. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of itsincircle). There are either one, two, or three of these for any given triangle.
Any line through the midpoint of a parallelogram bisects the area.
All area bisectors of a circle or other ellipse go through the center, and anychordsthrough the center bisect the area. In the case of a circle they are the diameters of the circle.
Optimization
editGiven a wire contour, the surface of least area spanning ( "filling" ) it is aminimal surface.Familiar examples includesoap bubbles.
The question of thefilling areaof theRiemannian circleremains open.[34]
The circle has the largest area of any two-dimensional object having the same perimeter.
Acyclic polygon(one inscribed in a circle) has the largest area of any polygon with a given number of sides of the same lengths.
A version of theisoperimetric inequalityfor triangles states that the triangle of greatest area among all those with a given perimeter isequilateral.[35]
The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.[36]
The ratio of the area of the incircle to the area of an equilateral triangle,,is larger than that of any non-equilateral triangle.[37]
The ratio of the area to the square of the perimeter of an equilateral triangle,is larger than that for any other triangle.[35]
See also
edit- Brahmagupta quadrilateral,a cyclic quadrilateral with integer sides, integer diagonals, and integer area.
- Equiareal map
- Heronian triangle,a triangle with integer sides and integer area.
- List of triangle inequalities
- One-seventh area triangle,an inner triangle with one-seventh the area of the reference triangle.
- Routh's theorem,a generalization of the one-seventh area triangle.
- Orders of magnitude—A list of areas by size.
- Derivation of the formula of a pentagon
- Planimeter,an instrument for measuring small areas, e.g. on maps.
- Area of a convex quadrilateral
- Robbins pentagon,a cyclic pentagon whose side lengths and area are all rational numbers.
References
edit- ^abcdefWeisstein, Eric W."Area".Wolfram MathWorld.Archivedfrom the original on 5 May 2012.Retrieved3 July2012.
- ^abcdefghi"Area Formulas".Math.com.Archivedfrom the original on 2 July 2012.Retrieved2 July2012.
- ^ab"Resolution 12 of the 11th meeting of the CGPM (1960)".Bureau International des Poids et Mesures.Archivedfrom the original on 2012-07-28.Retrieved15 July2012.
- ^Mark de Berg; Marc van Kreveld;Mark Overmars;Otfried Schwarzkopf (2000)."Chapter 3: Polygon Triangulation".Computational Geometry(2nd revised ed.).Springer-Verlag.pp.45–61.ISBN978-3-540-65620-3.
- ^Boyer, Carl B.(1959).A History of the Calculus and Its Conceptual Development.Dover.ISBN978-0-486-60509-8.
- ^abcWeisstein, Eric W."Surface Area".Wolfram MathWorld.Archivedfrom the original on 23 June 2012.Retrieved3 July2012.
- ^"Surface Area".CK-12 Foundation.Retrieved2018-10-09.
- ^abdo Carmo, Manfredo(1976).Differential Geometry of Curves and Surfaces.Prentice-Hall. p. 98,ISBN978-0-13-212589-5
- ^Walter Rudin (1966).Real and Complex Analysis,McGraw-Hill,ISBN0-07-100276-6.
- ^Gerald Folland (1999).Real Analysis: modern techniques and their applications,John Wiley & Sons, Inc., p. 20,ISBN0-471-31716-0
- ^Apostol, Tom (1967).Calculus.Vol. I: One-Variable Calculus, with an Introduction to Linear Algebra. pp. 58–59.ISBN9780471000051.
- ^Moise, Edwin (1963).Elementary Geometry from an Advanced Standpoint.Addison-Wesley Pub. Co.Retrieved15 July2012.
- ^abcdBureau international des poids et mesures (2006).The International System of Units (SI)(PDF).8th ed. Chapter 5.Archived(PDF)from the original on 2013-11-05.Retrieved2008-02-13.
- ^"Land Measurement Units in India: Standard Measurement Units, Land Conversion Table".Magicbricks Blog.2020-08-04.Retrieved2023-09-20.
- ^Mishra, Sunita (2023-06-13)."Land is measured in what units in India: All Types In 2023".Housing News.Retrieved2023-09-20.
- ^"Standard Land Measurement Units in India - Times Property".timesproperty.com.Retrieved2023-09-20.
- ^www.clicbrics.com."9 Land Measurement Units in India You Must Know - 2022".www.clicbrics.com.Retrieved2023-09-20.
- ^Heath, Thomas L.(2003).A Manual of Greek Mathematics.Courier Dover Publications. pp. 121–132.ISBN978-0-486-43231-1.Archivedfrom the original on 2016-05-01.
- ^Stewart, James (2003).Single variable calculus early transcendentals(5th. ed.). Toronto ON: Brook/Cole. p.3.ISBN978-0-534-39330-4.
However, by indirect reasoning, Eudoxus (fifth century B.C.) used exhaustion to prove the familiar formula for the area of a circle:
- ^Heath, Thomas L. (1921).A History of Greek Mathematics (Vol II).Oxford University Press. pp. 321–323.
- ^Weisstein, Eric W."Heron's Formula".MathWorld.
- ^"Euclid's Proof of the Pythagorean Theorem | Synaptic".Central College.Retrieved2023-07-12.
- ^Clark, Walter Eugene (1930).The Aryabhatiya of Aryabhata: An Ancient Indian Work on Mathematics and Astronomy(PDF).University of Chicago Press. p. 26.
- ^Xu, Wenwen; Yu, Ning (May 2013)."Bridge Named After the Mathematician Who Discovered the Chinese Remainder Theorem"(PDF).Notices of the American Mathematical Society.60(5): 596–597.
- ^Bourke, Paul (July 1988)."Calculating The Area And Centroid Of A Polygon"(PDF).Archived(PDF)from the original on 2012-09-16.Retrieved6 Feb2013.
- ^Averbach, Bonnie;Chein, Orin (2012).Problem Solving Through Recreational Mathematics.Dover. p.306.ISBN978-0-486-13174-0.
- ^Joshi, K. D. (2002).Calculus for Scientists and Engineers: An Analytical Approach.CRC Press. p. 43.ISBN978-0-8493-1319-6.Archivedfrom the original on 2016-05-05.
- ^abBraden, Bart (September 1986)."The Surveyor's Area Formula"(PDF).The College Mathematics Journal.17(4): 326–337.doi:10.2307/2686282.JSTOR2686282.Archived(PDF)from the original on 27 June 2012.Retrieved15 July2012.
- ^Trainin, J. (November 2007). "An elementary proof of Pick's theorem".Mathematical Gazette.91(522): 536–540.doi:10.1017/S0025557200182270.S2CID124831432.
- ^Matematika.PT Grafindo Media Pratama. pp. 51–.ISBN978-979-758-477-1.Archivedfrom the original on 2017-03-20.
- ^Get Success UN +SPMB Matematika.PT Grafindo Media Pratama. pp. 157–.ISBN978-602-00-0090-9.Archivedfrom the original on 2016-12-23.
- ^abcWeisstein, Eric W."Cone".Wolfram MathWorld.Archivedfrom the original on 21 June 2012.Retrieved6 July2012.
- ^ Mandelbrot, Benoît B. (1983).The fractal geometry of nature.Macmillan.ISBN978-0-7167-1186-5.Archivedfrom the original on 20 March 2017.Retrieved1 February2012.
- ^Gromov, Mikhael (1983)."Filling Riemannian manifolds".Journal of Differential Geometry.18(1): 1–147.CiteSeerX10.1.1.400.9154.doi:10.4310/jdg/1214509283.MR0697984.Archivedfrom the original on 2014-04-08.
- ^abChakerian, G.D. (1979) "A Distorted View of Geometry." Ch. 7 inMathematical Plums.R. Honsberger (ed.). Washington, DC: Mathematical Association of America, p. 147.
- ^Dorrie, Heinrich (1965),100 Great Problems of Elementary Mathematics,Dover Publ., pp. 379–380.
- ^Minda, D.; Phelps, S. (October 2008)."Triangles, ellipses, and cubic polynomials".American Mathematical Monthly.115(8): 679–689: Theorem 4.1.doi:10.1080/00029890.2008.11920581.JSTOR27642581.S2CID15049234.Archivedfrom the original on 2016-11-04.