Polygon Calculator

Name: Polygon - Geometry Calculator
Author: Jürgen Kummer

Calculations at a simple polygon. A polygon consists of straight edges and at least three vertices. It is simple when the edges don't intersect, so if the polygon isn't crossed. Here the edge lengths as well as the perimeter and area of the polygon can be calculated from the cartesian coordinates. First enter the number of vertices (3 to 30), then the x- and y-coordinate of each vertex. Choose the number of decimal places, then click Calculate. Side 1 runs from vertex 1 to vertex 2, side 2 from vertex 2 to 3, ..., the last side runs from vertex n to 1.

	Verices (n):	A simple polygon with 14 vertices.
	Vertex `1`: x= y=
	Vertex `2`: x= y=
	Vertex `3`: x= y=
	Vertex `4`: x= y=
	Vertex `5`: x= y=
	Vertex `6`: x= y=
	Vertex `7`: x= y=
	Vertex `8`: x= y=
	Vertex `9`: x= y=
	Vertex `10`: x= y=
	Vertex `11`: x= y=
	Vertex `12`: x= y=
	Vertex `13`: x= y=
	Vertex `14`: x= y=
	Vertex `15`: x= y=
	Vertex `16`: x= y=
	Vertex `17`: x= y=
	Vertex `18`: x= y=
	Vertex `19`: x= y=
	Vertex `20`: x= y=
	Vertex `21`: x= y=
	Vertex `22`: x= y=
	Vertex `23`: x= y=
	Vertex `24`: x= y=
	Vertex `25`: x= y=
	Vertex `26`: x= y=
	Vertex `27`: x= y=
	Vertex `28`: x= y=
	Vertex `29`: x= y=
	Vertex `30`: x= y=
	Round to decimal places.

	Edge 1:
	Edge 2:
	Edge 3:
	Edge 4:
	Edge 5:
	Edge 6:
	Edge 7:
	Edge 8:
	Edge 9:
	Edge 10:
	Edge 11:
	Edge 12:
	Edge 13:
	Edge 14:
	Edge 15:
	Edge 16:
	Edge 17:
	Edge 18:
	Edge 19:
	Edge 20:
	Edge 21:
	Edge 22:
	Edge 23:
	Edge 24:
	Edge 25:
	Edge 26:
	Edge 27:
	Edge 28:
	Edge 29:
	Edge 30:
	Perimeter (p):
Area (A):

Polygon shape. If this polygon is drawn crossed, then the upper area calculation is incorrect:

Formulas:

Length of edge i = \sqrt{{(x_{i + 1} - x_{i})}^{2} + {(y_{i + 1} - y_{i})}^{2}}

p = Σ_{i = 1}^{n} \sqrt{{(x_{i + 1} - x_{i})}^{2} + {(y_{i + 1} - y_{i})}^{2}}

A = \frac{| Σ_{i = 1}^{n} x_{i} \cdot y_{i + 1} - y_{i} \cdot x_{i + 1} |}{2}

with x_{n + 1} \to x_{1} and y_{n + 1} \to y_{1}

Σ is the sum symbol, | | is the absolute value.

x- and y-coordinate determine the position of a point right from and above the origin in the cartesian coordinate system. Edge lengths and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The calculation of the simple polygon is done using the Gaussian trapezoid formula, which was discovered by Carl Friedrich Gauss and Carl Gustav Jacob Jacobi in the 19th century. This formula assigns a trapezoid to each edge of the polygon. These trapezoids have positive or negative areas, depending on whether they are inside or outside the polygon. Finally, all of these areas are added together to obtain the area of the simple polygon. The perimeter is calculated in a similar way, only with the edge lengths instead of the areas.
The input is cumbersome, but any polygon can be calculated with this. For polygons with a specific shape, for which there are separate formulas and calculators, especially for triangles and quadrilaterals, these are definitely preferable.

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