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The best known and simplest formula is <math>T=bh/2,</math> where ''b'' is the [[length]] of the ''base'' of the triangle, and ''h'' is the ''height'' or ''[[Altitude (triangle)|altitude]]'' of the triangle. The term "base" denotes any side, and "height" denotes 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.<ref>{{Cite web |title=Euclid’s Proof of the Pythagorean Theorem {{!}} Synaptic |url=https://central.edu/writing-anthology/2019/01/31/159/ |access-date=2023-07-12 |website=Central College |language=en}}</ref> In 499 CE [[Aryabhata]], used this illustrated method in the ''[[Aryabhatiya]]'' (section 2.6).<ref>[https://archive.org/stream/The_Aryabhatiya_of_Aryabhata_Clark_1930#page/n1/mode/2up ''The Āryabhaṭīya'' by Āryabhaṭa] (translated into English by [[Walter Eugene Clark]], 1930) hosted online by the [[Internet Archive]].</ref> Although simple, this formula is only useful if the height can be readily found, which 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 integral]]s, [[Pick's theorem]], or other properties.<ref>{{MathWorld |title=Triangle area |urlname=TriangleArea}}</ref> ==History== [[Heron of Alexandria]] found what 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,<ref>{{cite book | author=Heath, Thomas L. | title=A History of Greek Mathematics (Vol II) | publisher=Oxford University Press | year=1921 | pages=321–323}}</ref> 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.<ref>{{MathWorld |urlname=HeronsFormula |title=Heron's Formula}}</ref> In 300 BCE Greek mathematician [[Euclid]] proved that the area of a triangle is half that of a parallelogram with the same base and height in his book ''Elements of Geometry''.<ref>{{Cite web |title=Euclid’s Proof of the Pythagorean Theorem {{!}} Synaptic |url=https://central.edu/writing-anthology/2019/01/31/159/ |access-date=2023-07-12 |website=Central College |language=en}}</ref> 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]]''.<ref>{{cite book|url=https://libarch.nmu.org.ua/bitstream/handle/GenofondUA/26818/f14e7857ef0bdd30ca0fd4ec057fe3c3.pdf|page=26|title=The Aryabhatiya of Aryabhata: An Ancient Indian Work on Mathematics and Astronomy|first=Walter Eugene|last=Clark|publisher=University of Chicago Press|year=1930}}</ref> A formula equivalent to Heron's was discovered by the Chinese independently of the Greeks. It was published in 1247 in ''Shushu Jiuzhang'' ("[[Mathematical Treatise in Nine Sections]]"), written by [[Qin Jiushao]].<ref>{{cite journal|url=http://www.ams.org/journals/notices/201305/rnoti-p596.pdf|title=Bridge Named After the Mathematician Who Discovered the Chinese Remainder Theorem|journal=Notices of the American Mathematical Society|last1=Xu|first1=Wenwen|last2=Yu|first2=Ning|pages=596–597|date=May 2013|volume=60|issue=5}}</ref> ==Using trigonometry== [[File:Triangle.TrigArea.svg|frame|Applying trigonometry to find the altitude ''h''.]] The height of a triangle can be found through the application of [[trigonometry]]. ; Knowing SAS (side-angle-side): Using the labels in the image on the right, the altitude is {{nowrap|''h'' {{=}} ''a'' sin <math>\gamma</math>}}. Substituting this in the formula <math>T=\tfrac12 bh</math> derived above, the area of the triangle can be expressed as: :<math>T = \tfrac12 ab\sin \gamma = \tfrac12 bc\sin \alpha = \tfrac12 ca\sin \beta</math> (where α is the interior angle at ''A'', β is the interior angle at ''B'', <math>\gamma</math> is the interior angle at ''C'' and ''c'' is the line '''AB'''). Furthermore, since sin α = sin (''π'' − α) = sin (β + <math>\gamma</math>), and similarly for the other two angles: :<math>T = \tfrac12 ab\sin (\alpha+\beta) = \tfrac12 bc\sin (\beta+\gamma) = \tfrac12 ca\sin (\gamma+\alpha).</math> ; Knowing AAS (angle-angle-side): :<math>T = \frac {b^{2}(\sin \alpha)(\sin (\alpha + \beta))}{2\sin \beta},</math> and analogously if the known side is ''a'' or ''c''. ; Knowing ASA (angle-side-angle): :<math>T = \frac{a^{2}}{2(\cot \beta + \cot \gamma)} = \frac{a^{2} (\sin \beta)(\sin \gamma)}{2\sin(\beta + \gamma)},</math> and analogously if the known side is ''b'' or ''c''.<ref name=":1">{{MathWorld |title=Triangle |urlname=Triangle}}</ref> ==Using side lengths (Heron's formula) {{anchor|Heron's formula}}== {{main|Heron's formula}} 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]], :<math>T = \sqrt{s(s-a)(s-b)(s-c)}</math> where <math display=inline>s= \tfrac12(a+b+c)</math> is the [[semiperimeter]], or half of the triangle's perimeter. Three other equivalent ways of writing Heron's formula are :<math>\begin{align} T &= \tfrac14 \sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)} \\[5mu] &= \tfrac14 \sqrt{2(a^2b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)} \\[5mu] &= \tfrac14 \sqrt{(a+b-c) (a-b+c) (-a+b+c) (a+b+c)}. \end{align}</math> ===Formulas resembling Heron's formula=== 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<sub>a</sub>'', ''m<sub>b</sub>'', and ''m<sub>c</sub>'' and their semi-sum {{nowrap|(''m<sub>a</sub>'' + ''m<sub>b</sub>'' + ''m<sub>c</sub>'')/2}} as σ, we have<ref>Benyi, Arpad, "A Heron-type formula for the triangle," ''Mathematical Gazette'' 87, July 2003, 324–326.</ref> :<math>T = \tfrac43 \sqrt{\sigma (\sigma - m_a)(\sigma - m_b)(\sigma - m_c)}.</math> Next, denoting the altitudes from sides ''a'', ''b'', and ''c'' respectively as ''h<sub>a</sub>'', ''h<sub>b</sub>'', and ''h<sub>c</sub>'', and denoting the semi-sum of the reciprocals of the altitudes as <math>H = (h_a^{-1} + h_b^{-1} + h_c^{-1})/2</math> we have<ref>Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle," ''Mathematical Gazette'' 89, November 2005, 494.</ref> :<math>T^{-1} = 4 \sqrt{H(H-h_a^{-1})(H-h_b^{-1})(H-h_c^{-1})}.</math> And denoting the semi-sum of the angles' sines as {{nowrap|''S'' {{=}} [(sin α) + (sin β) + (sin γ)]/2}}, we have<ref>Mitchell, Douglas W., "A Heron-type area formula in terms of sines," ''Mathematical Gazette'' 93, March 2009, 108–109.</ref> :<math>T = D^{2} \sqrt{S(S-\sin \alpha)(S-\sin \beta)(S-\sin \gamma)}</math> where ''D'' is the diameter of the [[circumcircle]]: <math>D=\tfrac{a}{\sin \alpha} = \tfrac{b}{\sin \beta} = \tfrac{c}{\sin \gamma}.</math> ==Using vectors== The area of triangle ABC is half of the area of a [[parallelogram]]: :<math> T = \tfrac12\bigl\|(\mathbf b - \mathbf a) \wedge (\mathbf c - \mathbf a)\bigr\| = \tfrac12\bigl\|\mathbf a \wedge \mathbf b + \mathbf b \wedge \mathbf c + \mathbf c \wedge \mathbf a \bigr\|, </math> where {{tmath|\mathbf a}}, {{tmath|\mathbf b}}, and {{tmath|\mathbf c}} are [[Vector (geometric)|vectors]] to the triangle's vertices from any arbitrary origin point, so that {{tmath|\mathbf b - \mathbf a}} and {{tmath|\mathbf c - \mathbf a}} are the [[displacement vector|translation vectors]] from vertex {{tmath|A}} to each of the others, and {{tmath|\wedge}} is the [[wedge product]]. If vertex {{tmath|A}} is taken to be the origin, this simplifies to {{nobr|<math>\tfrac12\| \mathbf b \wedge \mathbf c \|</math>}}. The oriented relative [[Parallelogram#Area|area of a parallelogram]] in any affine space, a type of [[bivector]], is defined as {{tmath|\mathbf u \wedge \mathbf v}} where {{tmath|\mathbf u}} and {{tmath|\mathbf v}} are translation vectors from one vertex of 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, 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 product]]s. Taking vertex {{tmath|A}} to be the origin and calling translation vectors to the other vertices {{tmath|\mathbf b}} and {{tmath|\mathbf c}}, :<math>T = \tfrac12 \sqrt{\mathbf{b}^2 \mathbf{c}^2 - (\mathbf{b} \cdot \mathbf{c})^2},</math> where for any Euclidean vector {{nobr|<math>\mathbf v^2 = \|\mathbf v\|^2 = \mathbf v \cdot \mathbf v</math>}}.<ref>The quantity {{tmath|\mathbf v^2}} represents [[geometric product]] of a vector {{tmath|\mathbf v}} with itself.</ref> This area formula can be derived from the previous one using the elementary vector identity {{nobr|<math> \mathbf u^2 \mathbf v^2 = (\mathbf u \cdot \mathbf v)^2 + \|\mathbf u \wedge \mathbf v \|^2</math>.}} In two-dimensional Euclidean space, for a vector {{tmath|\mathbf b}} with coordinates {{tmath|(x_B, y_B)}} and vector {{tmath|\mathbf c}} with coordinates {{tmath|(x_C, y_C)}}, the magnitude of the wedge product is :<math>\| \mathbf b \wedge \mathbf c \| = |x_B y_C - x_C y_B|.</math> (See the following section.) ==Using coordinates== If vertex ''A'' is located at the origin (0,&nbsp;0) of a [[Cartesian coordinate system]] and the coordinates of the other two vertices are given by {{nowrap|''B'' {{=}} (''x<sub>B</sub>'', ''y<sub>B</sub>'')}} and {{nowrap|''C'' {{=}} (''x<sub>C</sub>'', ''y<sub>C</sub>'')}}, then the area can be computed as {{frac|1|2}} times the [[absolute value]] of the [[determinant]] :<math>T = \tfrac12\left|\det\begin{pmatrix}x_B & x_C \\ y_B & y_C \end{pmatrix}\right| = \tfrac12 |x_B y_C - x_C y_B|.</math> For three general vertices, the equation is: :<math>T = \tfrac12 \left| \det\begin{pmatrix}x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1\end{pmatrix} \right| = \tfrac12 \big| x_A y_B - x_A y_C + x_B y_C - x_B y_A + x_C y_A - x_C y_B \big|,</math> which can be written as :<math>T = \tfrac12 \big| (x_A - x_C) (y_B - y_A) - (x_A - x_B) (y_C - y_A) \big|.</math> If the points are labeled sequentially in the counterclockwise direction, the above determinant expressions are positive and the absolute value signs can be omitted.<ref>{{cite journal |author=Bart Braden |title=The Surveyor's Area Formula |journal=The College Mathematics Journal |volume=17 |issue=4 |year=1986 |pages=326–337 |url=http://www.maa.org/pubs/Calc_articles/ma063.pdf |doi=10.2307/2686282 |jstor=2686282 |access-date=5 January 2012 |archive-date=5 November 2003 |archive-url=https://web.archive.org/web/20031105063724/http://www.maa.org/pubs/Calc_articles/ma063.pdf |url-status=dead }}</ref> 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 {{nowrap|''a'' {{=}} ''x<sub>A</sub>'' + ''y<sub>A</sub>i''}}, {{nowrap|''b'' {{=}} ''x<sub>B</sub>'' + ''y<sub>B</sub>i''}}, and {{nowrap|''c'' {{=}} ''x<sub>C</sub>'' + ''y<sub>C</sub>i''}}, and denote their complex conjugates as <math>\bar a</math>, <math>\bar b</math>, and <math>\bar c</math>, then the formula :<math>T=\frac{i}{4}\begin{vmatrix}a & \bar a & 1 \\ b & \bar b & 1 \\ c & \bar c & 1 \end{vmatrix}</math> is equivalent to the shoelace formula. In three dimensions, the area of a general triangle {{nowrap|''A'' {{=}} (''x<sub>A</sub>'', ''y<sub>A</sub>'', ''z<sub>A</sub>'')}}, {{nowrap|''B'' {{=}} (''x<sub>B</sub>'', ''y<sub>B</sub>'', ''z<sub>B</sub>'')}} and {{nowrap|''C'' {{=}} (''x<sub>C</sub>'', ''y<sub>C</sub>'', ''z<sub>C</sub>''}}) 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): :<math>T = \tfrac12 \sqrt{\begin{vmatrix} x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1 \end{vmatrix}^2 + \begin{vmatrix} y_A & y_B & y_C \\ z_A & z_B & z_C \\ 1 & 1 & 1 \end{vmatrix}^2 + \begin{vmatrix} z_A & z_B & z_C \\ x_A & x_B & x_C \\ 1 & 1 & 1 \end{vmatrix}^2 }.</math> ==Using line integrals== 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 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<sub>i</sub>'',''y<sub>i</sub>'') and (''x''<sub>''i''+1</sub>,''y''<sub>''i''+1</sub>) is given by the base times the mean height, namely {{nowrap|(''x''<sub>''i''+1</sub> − ''x<sub>i</sub>'')(''y<sub>i</sub>'' + ''y''<sub>''i''+1</sub>)/2}}. The sign of 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, 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. {{nowrap|''x''<sub>''i''+1</sub> − ''x<sub>i</sub>''}} 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<sub>i</sub>'',θ<sub>''i''</sub>) and (''r''<sub>''i''+1</sub>,θ<sub>''i''+1</sub>) of a polygon is given directly by {{nowrap|''r<sub>i</sub>r''<sub>''i''+1</sub>sin(θ<sub>''i''+1</sub> − θ<sub>''i''</sub>)/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 ({{nowrap|''x'' {{=}} 0}}) is immaterial for line integration in cartesian coordinates, so is the choice of zero heading ({{nowrap|θ {{=}} 0}}) immaterial here. =={{anchor|Using Pick's Theorem}}Using Pick's theorem== See [[Pick's theorem]] for a technique for finding the area of any arbitrary [[Lattice graph|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: :<math>T = I + \tfrac12 B - 1</math> where ''<math>I</math>'' is the number of internal lattice points and ''B'' is the number of lattice points lying on the border of the polygon. ==Other area formulas== Numerous other area formulas exist, such as :<math>T = r \cdot s,</math> where ''r'' is the [[inradius]], and ''s'' is the [[semiperimeter]] (in fact, this formula holds for ''all'' [[tangential polygon]]s), and<ref name=Kiss>{{cite web| url = http://forumgeom.fau.edu/FG2016volume16/FG201634.pdf| title = Sa ́ndor Nagydobai Kiss, "A Distance Property of the Feuerbach Point and Its Extension", ''Forum Geometricorum'' 16, 2016, 283–290.}}</ref>{{rp|Lemma 2}} :<math>T=r_a(s-a)=r_b(s-b)=r_c(s-c)</math> where <math>r_a, \, r_b,\, r_c</math> are the radii of the [[excircles]] tangent to sides ''a, b, c'' respectively. We also have :<math>T = \tfrac12 D^{2}(\sin \alpha)(\sin \beta)(\sin \gamma)</math> and<ref>{{cite web|title=Circumradius |url=http://www.artofproblemsolving.com/Wiki/index.php/Circumradius |website=AoPSWiki |access-date=26 July 2012 |url-status=dead |archive-url=https://web.archive.org/web/20130620011530/http://www.artofproblemsolving.com/Wiki/index.php/Circumradius |archive-date=20 June 2013 }}</ref> :<math>T = \frac{abc}{2D} = \frac{abc}{4R}</math> for circumdiameter ''D''; and<ref>Mitchell, Douglas W., "The area of a quadrilateral," ''Mathematical Gazette'' 93, July 2009, 306–309.</ref> :<math>T = \tfrac14(\tan \alpha)(b^{2}+c^{2}-a^{2})</math> for angle α ≠ 90°. The area can also be expressed as<ref>Pathan, Alex, and Tony Collyer, "Area properties of triangles revisited," ''[[Mathematical Gazette]]'' 89, November 2005, 495–497.</ref> :<math>T = \sqrt{rr_ar_br_c}.</math> In 1885, Baker<ref>Baker, Marcus, "A collection of formulae for the area of a plane triangle," ''Annals of Mathematics'', part 1 in vol. 1(6), January 1885, 134–138; part 2 in vol. 2(1), September 1885, 11–18. The formulas given here are #9, #39a, #39b, #42, and #49. The reader is advised that several of the formulas in this source are not correct.</ref> gave a collection of over a hundred distinct area formulas for the triangle. These include: :<math>T = \tfrac12\sqrt[3]{abch_ah_bh_c},</math> :<math>T = \tfrac12 \sqrt{abh_ah_b},</math> :<math>T = \frac{a+b}{2(h_a^{-1} + h_b^{-1})},</math> :<math>T = \frac{Rh_bh_c}{a}</math> for circumradius (radius of the circumcircle) ''R'', and :<math>T = \frac{h_ah_b}{2 \sin \gamma}.</math> ==Upper bound on the area== The area ''T'' of any triangle with perimeter ''p'' satisfies :<math>T\le \tfrac{p^2}{12\sqrt{3}},</math> with equality holding if and only if the triangle is equilateral.<ref>Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.</ref><ref>Rosenberg, Steven; Spillane, Michael; and Wulf, Daniel B. "Heron triangles and moduli spaces", ''Mathematics Teacher'' 101, May 2008, 656–663.</ref>{{rp|657}} Other upper bounds on the area ''T'' are given by<ref>Posamentier, Alfred S., and Lehmann, Ingmar, ''[[The Secrets of Triangles]]'', Prometheus Books, 2012.</ref>{{rp|p.290}} :<math>4\sqrt{3}T \leq a^2+b^2+c^2</math> and :<math>4\sqrt{3}T \leq \frac{9abc}{a+b+c}, </math> both again holding if and only if the triangle is equilateral. ==Bisecting the area== There are infinitely many [[Bisection#Area bisectors and perimeter bisectors|lines that bisect the area of a triangle]].<ref name=Dunn>Dunn, J.A., and Pretty, J.E., "Halving a triangle," ''[[Mathematical Gazette]]'' 56, May 1972, 105–108.</ref> Three of them are the medians, which are the only area bisectors that go through the centroid. Three other area bisectors are parallel to the triangle's sides. Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter. There can be one, two, or three of these for any given triangle. ==See also== *[[Area of a circle]] *[[Congruence (geometry)#Congruence of triangles|Congruence of triangles]] ==References== {{reflist}} [[Category:Area]] [[Category:Triangles]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages transcluded onto the current version of this page<span class="posteditwindowhelplinks"> (<a href="/wiki/Help:Transclusion" title="Help:Transclusion">help</a>)</span>:</span> </p></div><ul> <li><a href="/wiki/Template:Anchor" title="Template:Anchor">Template:Anchor</a> (<a href="/w/index.php?title=Template:Anchor&action=edit" title="Template:Anchor">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Cite_book" title="Template:Cite book">Template:Cite book</a> (<a href="/w/index.php?title=Template:Cite_book&action=edit" title="Template:Cite book">view source</a>) (protected)</li><li><a 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