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The area formula can also be applied to self-overlapping polygons since the meaning of area is still clear even though self-overlapping polygons are not generally simple. [6] Furthermore, a self-overlapping polygon can have multiple "interpretations" but the Shoelace formula can be used to show that the polygon's area is the same regardless of ...
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.
Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. [4] For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus .
In either case, the area formula is correct in absolute value. This is commonly called the shoelace formula or surveyor's formula. [6] The area A of a simple polygon can also be computed if the lengths of the sides, a 1, a 2, ..., a n and the exterior angles, θ 1, θ 2, ..., θ n are known, from:
In geometry a quadrilateral is a four-sided polygon, ... Irregular quadrilateral ... Another area formula in terms of the sides and angles, ...
The second moment of area about the origin for any simple polygon on the XY-plane can be computed in general by summing contributions from each segment of the polygon after dividing the area into a set of triangles. This formula is related to the shoelace formula and can be considered a special case of Green's theorem. A polygon is assumed to ...
The quotients formed by the area of these polygons divided by the square of the circle radius can be made arbitrarily close to π as the number of polygon sides becomes large, proving that the area inside the circle of radius r is πr 2, π being defined as the ratio of the circumference to the diameter (C/d).
Simple formulae are known for computing the area of the interior of a polygon. These include the shoelace formula for arbitrary polygons, [21] and Pick's theorem for polygons with integer vertex coordinates. [12] [22]