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Combining these area computations gives Pick's formula for triangles, and combining triangles gives Pick's formula for arbitrary polygons. [7] [8] [13] Alternatively, instead of using grid squares centered on the grid points, it is possible to use grid squares having their vertices at the grid points.
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]
Overlaying the "hypotenuses" from both figures results in a very thin parallelogram (represented with the four red dots in the above image) with an area of exactly one grid square (Pick's theorem gives 0 [1] + 4 [2] / 2 − 1 = 1), which corresponds to the "missing" area.
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
Area is the measure of a region's size on a surface. The area of a plane region or plane area refers to the area of a shape or planar lamina, while surface area refers to the area of an open surface or the boundary of a three-dimensional object.
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The dot planimeter is physical device for estimating the area of shapes based on the same principle. It consists of a square grid of dots, printed on a transparent sheet; the area of a shape can be estimated as the product of the number of dots in the shape with the area of a grid square. [8]
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