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Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + 96 / 2 − 1 = 48 [1]. In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary.
Area#Area formulas – Size of a two-dimensional surface; Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities
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 a = x A + y A i , b = x B + y B i , and c = x C + y C i , and denote their complex conjugates as a ¯ {\displaystyle {\bar {a}}} , b ¯ {\displaystyle {\bar {b}}} , and c ...
The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when =, = /, and the altitude of the triangle from the base of length is equal to . The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is 2 2 / 3 {\displaystyle 2 ...
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.