section_path: Properties and formulas/Area/Simple polygons
section_name: Simple polygons
paragraph_text: Simple polygons
  
  If the polygon is non-self-intersecting (that is, simple), the signed area is
  
  or, using determinants
  
  where Q i, j {\displaystyle Q_{i,j}} is the squared distance between (x i, y i) {\displaystyle (x_{i},y_{i})} and (x j, y j). {\displaystyle (x_{j},y_{j}).}
  
  The signed area depends on the ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive x -axis to the positive y -axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in absolute value. This is commonly called the shoelace formula or surveyor's formula.
  
  The area A of a simple polygon can also be computed if the lengths of the sides, a, a,..., a and the exterior angles, θ, θ,..., θ are known, from:
  
  The formula was described by Lopshits in 1963.
  
  If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1.
  
  In every polygon with perimeter p and area A, the isoperimetric inequality p 2 > 4 π A {\displaystyle p^{2}>4\pi A} holds.
  
  For any two simple polygons of equal area, the Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon.
  
  The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides do determine the area. Of all n -gons with given side lengths, the one with the largest area is cyclic. Of all n -gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).