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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]
The area can be also expressed in terms of bimedians as [16] = , where the lengths of the bimedians are m and n and the angle between them is φ. Bretschneider's formula [17] [14] expresses the area in terms of the sides and two opposite angles:
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.
If an angle and its two included sides are given, the area is where C is the given angle and a and b are its included sides. [2] If the triangle is graphed on a coordinate plane, a matrix can be used and is simplified to the absolute value of 1 2 ( x 1 y 2 + x 2 y 3 + x 3 y 1 − x 2 y 1 − x 3 y 2 − x 1 y 3 ) {\displaystyle {\tfrac {1 ...
Coast – Area where land meets the sea or ocean; Coastal plain – Area of flat, low-lying land adjacent to a seacoast; Col – Lowest point on a mountain ridge between two peaks; Complex crater – Large impact craters with uplifted centres; Complex volcano – Landform of more than one related volcanic centre
The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram. The base × height area formula can also be derived using the figure to the right. The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the ...
The metes and bounds system was used to describe a town of a generally rectangular shape, 4 to 6 miles (6.4 to 9.7 km) on a side. Within this boundary, a map or plat was maintained that showed all the individual lots or properties. There are some difficulties with this system: Irregular shapes for properties make for much more complex descriptions.
If the incircle is tangent to the sides AB, BC, CD, DA at T 1, T 2, T 3, T 4 respectively, and if N 1, N 2, N 3, N 4 are the isotomic conjugates of these points with respect to the corresponding sides (that is, AT 1 = BN 1 and so on), then the Nagel point of the tangential quadrilateral is defined as the intersection of the lines N 1 N 3 and N ...