Editing Triangle (section)

=== Area ===
{{main article|Area of a triangle}}
[[File:Triangle.GeometryArea.svg|upright=1.55|thumb|The area formula for a triangle can be proven by cutting two copies of the triangle into pieces and rearranging them into a rectangle.]]
In the Euclidean plane, [[area]] is defined by comparison with a square of side length {{tmath|1}}, which has area 1. There are several ways to calculate the area of an arbitrary triangle. One of the oldest and simplest is to take half the product of the length of one side {{tmath|b}} (the base) times the corresponding altitude {{tmath|h}}:{{sfn|Ryan|2008|p=[https://books.google.com/books?id=b_qM4HImlPgC&pg=PA98 98]}}
<math display="block"> T = \tfrac{1}{2}bh. </math>

This formula can be proven by cutting up the triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base {{tmath|b}} and height {{tmath|h}}.

[[File:Triangle.TrigArea.svg|thumb|right|upright=0.8|Applying trigonometry to find the altitude {{math|1=''h''}}]]
If two sides {{tmath|a}} and {{tmath|b}} and their included angle <math> \gamma </math> are known, then the altitude can be calculated using trigonometry, {{tmath|1= h = a \sin(\gamma)}}, so the area of the triangle is:
<math display="block"> T = \tfrac{1}{2}ab \sin \gamma. </math>

[[Heron's formula]], named after [[Heron of Alexandria]], is a formula for finding the area of a triangle from the lengths of its sides <math> a </math>, <math> b </math>, <math> c </math>. Letting <math> s = \tfrac12(a + b + c) </math> be the  [[semiperimeter]],<ref>{{MacTutor|id=Heron |title=Heron of Alexandria}}</ref>
<math display="block"> T = \sqrt{s(s - a)(s - b)(s - c)}. </math>

[[File:Lexell's theorem in the plane.png|thumb|Orange triangles {{math|△''ABC''}} share a base {{mvar|AB}} and area. The locus of their apex {{mvar|C}} is a line (dashed green) parallel to the base. This is the Euclidean version of [[Lexell's theorem]].]]
Because the ratios between areas of shapes in the same plane are preserved by [[affine transformation]]s, the relative areas of triangles in any [[affine plane]] can be defined without reference to a notion of distance or squares. In any affine space (including Euclidean planes), every triangle with the same base and [[signed area|oriented area]] has its apex (the third vertex) on a line parallel to the base, and their common area is half of that of a [[parallelogram]] with the same base whose opposite side lies on the parallel line. This affine approach was developed in Book 1 of Euclid's ''Elements''.{{sfn|Heath|1926|loc=Propositions 36–41}}

Given [[affine coordinates]] (such as [[Cartesian coordinates]]) {{tmath|(x_A, y_A)}}, {{tmath|(x_B, y_B)}}, {{tmath|(x_C, y_C)}} for the vertices of a triangle, its relative oriented area can be calculated using the [[shoelace formula]],

<math display=block>\begin{align}
T &= \tfrac12 \begin{vmatrix}x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1\end{vmatrix}
= \tfrac12 \begin{vmatrix} x_A & x_B \\ y_A & y_B \end{vmatrix}
+ \tfrac12 \begin{vmatrix} x_B & x_C \\ y_B & y_C \end{vmatrix}
+ \tfrac12 \begin{vmatrix} x_C & x_A \\ y_C & y_A \end{vmatrix} \\
&= \tfrac12(x_Ay_B - x_By_A + x_By_C - x_Cy_B + x_Cy_A - x_Ay_C),
\end{align}</math>

where <math>| \cdot |</math> is the [[matrix determinant]].<ref>{{cite journal |first=Bart |last=Braden |title=The Surveyor's Area Formula |journal=The College Mathematics Journal |volume=17 |issue=4 |year=1986 |pages=326–337 |url=https://www.maa.org/sites/default/files/pdf/pubs/Calc_Articles/ma063.pdf |doi=10.2307/2686282 |jstor=2686282 |archive-url=https://web.archive.org/web/20140629065751/https://www.maa.org/sites/default/files/pdf/pubs/Calc_Articles/ma063.pdf |archive-date=29 June 2014 |url-status=dead}}</ref>