section_path: Properties/Area
section_name: Area
paragraph_text: Area
  
  In the Euclidean plane, area is defined by comparison with a square of side length 1 {\displaystyle 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 b {\displaystyle b} (the base) times the corresponding altitude h {\displaystyle h}: T = 1 2 b h. {\displaystyle T={\tfrac {1}{2}}bh.}
  
  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 b {\displaystyle b} and height h {\displaystyle h}.
  
  If two sides a {\displaystyle a} and b {\displaystyle b} and their included angle γ {\displaystyle \gamma } are known, then the altitude can be calculated using trigonometry, h = a sin (γ) {\displaystyle h=a\sin(\gamma)}, so the area of the triangle is: T = 1 2 a b sin γ. {\displaystyle T={\tfrac {1}{2}}ab\sin \gamma.}
  
  Heron's formula, named after Heron of Alexandria, is a formula for finding the area of a triangle from the lengths of its sides a {\displaystyle a}, b {\displaystyle b}, c {\displaystyle c}. Letting s = 1 2 (a + b + c) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} be the semiperimeter, T = s (s − a) (s − b) (s − c). {\displaystyle T={\sqrt {s(s-a)(s-b)(s-c)}}.}
  
  Because the ratios between areas of shapes in the same plane are preserved by affine transformations, 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 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.
  
  Given affine coordinates (such as Cartesian coordinates) (x A, y A) {\displaystyle (x_{A},y_{A})}, (x B, y B) {\displaystyle (x_{B},y_{B})}, (x C, y C) {\displaystyle (x_{C},y_{C})} for the vertices of a triangle, its relative oriented area can be calculated using the shoelace formula,
  
  T = 1 2 | x A x B x C y A y B y C 1 1 1 | = 1 2 | x A x B y A y B | + 1 2 | x B x C y B y C | + 1 2 | x C x A y C y A | = 1 2 (x A y B − x B y A + x B y C − x C y B + x C y A − x A y C), {\displaystyle {\begin{aligned}T&={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{vmatrix}}={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}\\y_{A}&y_{B}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{C}&x_{A}\\y_{C}&y_{A}\end{vmatrix}}\\&={\tfrac {1}{2}}(x_{A}y_{B}-x_{B}y_{A}+x_{B}y_{C}-x_{C}y_{B}+x_{C}y_{A}-x_{A}y_{C}),\end{aligned}}}
  
  where | ⋅ | {\displaystyle |\cdot |} is the matrix determinant.