# Polygon

Different views of polygons and polygonal areas

A polygon (from ancient Greek πολυγώνιον polygṓnion 'polygon'; from πολύς polýs 'much' and γωνία gōnía 'angle') or polygon is a flat geometric figure in elementary geometry that is formed by a closed line .

A polygon is a two-dimensional polytope .

A polygon is obtained by (non-collinear) in a plane at least three different points by stretching are bonded together. This creates a closed line ( polygon ) with as many corners , for example a triangle (3 points, 3 lines) or a square (4 points, 4 lines).

The enclosed area is often referred to as a polygon, as in planimetry .

## Definition and terms

A polygon is a figure defined by a tuple of different points. ${\ displaystyle P: = \ left (P_ {1}, P_ {2}, \ dotsc, P_ {n} \ right), P_ {i} \ in \ mathbb {R} ^ {2}, 1 \ leq i \ leq n}$${\ displaystyle n}$

• The points are called the corner points or corners of the polygon for short , a polygon with corners is called -Eck or (especially in English literature) also -on.${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$
• The lines and are called the sides of the polygon.${\ displaystyle {\ overline {P_ {i} P_ {i + 1}}} \ left (i = 1, \ dotsc, n-1 \ right)}$${\ displaystyle {\ overline {P_ {n} P_ {1}}}}$
• All connecting lines between two corner points that are not sides are called diagonals .

Sometimes further conditions are required for the definition of a polygon, but they are not formally necessary:

• A polygon has at least three corner points that differ from one another in pairs. That excludes a "two-corner".
• Three adjacent corner points are not on a straight line. Also , , and , , will be considered as adjacent vertices. This excludes corners with straight angles.${\ displaystyle P_ {n}}$${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$${\ displaystyle P_ {n-1}}$${\ displaystyle P_ {n}}$${\ displaystyle P_ {1}}$

## classification

Historical illustration of polygons (1699)

### By number of corners

Polygons are typically named after the number of corners (weight of the polygon).

### Regular polygon

If a polygon has the same sides and the same interior angles, then it is called a regular polygon or a regular polygon. Many regular polygons can be constructed with compasses and rulers ( constructible polygons ).
Z + L means: can be constructed with a compass and ruler.

List of regular polygons
Corners designation Latin Z + L Specialty
1 One corner Monogonic - Point
2 Delta Digon - Straight
3 triangle Trine Yes 1. Fermat prime number ${\ displaystyle 3 = 2 ^ {2 ^ {0}} + 1}$
4th square Tetragon Yes square
5 pentagon Pentagon Yes 2. Fermat prime number ${\ displaystyle 5 = 2 ^ {2 ^ {1}} + 1}$
6th hexagon hexagon Yes
7th heptagon heptagon No Approximate construction possible
8th octagon Octagon Yes english oct a gon
9 Neuneck Nonagon No rarer Enneagon , approximation construction possible
10 decagon Decagon Yes
11 Elf Hendekagon No Approximate construction possible
12 Dodecagon Dodecagon Yes
13 Triangle Tridecagon No
17th Seventeenth corner Heptadecagon Yes 3. Fermat prime number ${\ displaystyle 17 = 2 ^ {2 ^ {2}} + 1}$
18th Eighteenth Octodecagon No english oct a decagon, octakaidecagon
20th Twentieth Ikosagon Yes
21st Twenty One Ikosihenagon No
24 Twenty-four square Icositetragon Yes
30th Thirty-corner Triakontagon Yes
40 Tetragonal Tetracontagon Yes
50 Fifty-point Pentacontagon No
51 Fifty-one Pentacontahenagon Yes
60 Hexagon Hexagon contact Yes
70 Seventieth Heptakontagon No
80 Octagon Octocontagon Yes english oct a contagon
85 Eighty-five square Octocontapentagon Yes english oct a contapentagon
90 Ninety-four Enneakontagon No
100 Hunderteck Hectogon No
257 257 corner Yes 4. Fermat's prime number ${\ displaystyle 257 = 2 ^ {2 ^ {3}} + 1}$
1,000 Thousands of pieces Chiliagon
10,000 Tens of thousands Myriagon
65,537 65537-corner Yes 5. Fermat's prime number ${\ displaystyle 65,537 = 2 ^ {2 ^ {4}} + 1}$
100,000 Hundreds of thousands
1,000,000 1000000 corner Megagon
4,294,967,295 4294967295-corner Yes Largest known odd number of corners that can theoretically be constructed with a compass and ruler
${\ displaystyle 10 ^ {100}}$ Googoleck Googolgon Corner number: a 1 with 100 zeros

### More types

Classification of polygons
Overturned polygon
If the edges intersect (touch) not only at the corner points, the polygon is referred to as overturning . If there is no self-intersection, the polygon is called simple .
Non-skipping polygon
Polygons that are not overturned can be convex (all interior angles are smaller than 180 °) or non- convex (at least one interior angle is greater than 180 °).
Planar polygon
In the plane (planar) polygon.
Non-planar polygon
In space (non-planar) polygon.

Polygons can be equilateral or equiangular :

Regular polygon
If a polygon has the same sides as well as the same interior angles, then it is called a regular polygon or a regular polygon.
Star polygon
Planar overturned regular polygons are also known as star polygons because of their appearance .
Orthogonal polygon
With orthogonal polygons, all edges meet at right angles (that is, the interior angle is either 90 ° or 270 ° at each edge).

## properties

### angle

In a flat corner that is not overturned is the sum of the interior angles${\ displaystyle n}$

${\ displaystyle \ alpha _ {1} + \ dotsb + \ alpha _ {n} = (n-2) \ cdot 180 ^ {\ circ}}$.

The following then applies to the sum of the outer angles regardless of the number of corners

${\ displaystyle \ alpha _ {1} '+ \ dotsb + \ alpha _ {n}' = 360 ^ {\ circ}}$.

In addition, if all the inside and outside angles are the same, then these have the value

${\ displaystyle \ alpha = {\ frac {(n-2)} {n}} \ cdot 180 ^ {\ circ}}$   or   .${\ displaystyle \ alpha '= {\ frac {1} {n}} \ cdot 360 ^ {\ circ}}$

### Diagonals

For polygons that are not crossed over, the following considerations apply when calculating the number of diagonals:

1. Each of the corners can be connected to one of the other corners by a link.${\ displaystyle n}$
2. The connection from corner to corner is identical to the connection from to .${\ displaystyle P_ {a}}$${\ displaystyle P_ {b}}$${\ displaystyle P_ {b}}$${\ displaystyle P_ {a}}$
3. Exactly connections are sides of the polygon.${\ displaystyle n}$

So a corner that is not overturned has exactly diagonals. In the case of a non-convex polygon, there are diagonals outside the polygon (in the area of ​​a truncated interior angle). ${\ displaystyle n}$${\ displaystyle {\ tfrac {n \ cdot (n-1)} {2}} - n = {\ tfrac {n \ cdot (n-3)} {2}}}$

### scope

If the corner points of a flat simple polygon are given by Cartesian coordinates , the perimeter of the polygon can be determined by adding the side lengths calculated using the Pythagorean theorem : ${\ displaystyle (x_ {i}, y_ {i})}$

${\ displaystyle \ mathrm {U} \ = \ {\ sqrt {(x_ {1} -x_ {n}) ^ {2} + (y_ {1} -y_ {n}) ^ {2}}} \ + \ \ sum _ {i = 1} ^ {n-1} {\ sqrt {(x_ {i + 1} -x_ {i}) ^ {2} + (y_ {i + 1} -y_ {i}) ^ {2}}}}$

### surface

If the corner points of a flat simple polygon are given by Cartesian coordinates , the area of ​​the polygon can be calculated according to the Gaussian trapezoidal formula: ${\ displaystyle (x_ {i}, y_ {i})}$

${\ displaystyle \ mathrm {2} A \ = \ \ left | \ sum _ {i = 1} ^ {n} (y_ {i} + y_ {i + 1}) \ cdot (x_ {i} -x_ { i + 1}) \ right | \ = \ \ left | \ sum _ {i = 1} ^ {n} (x_ {i} + x_ {i + 1}) \ cdot (y_ {i + 1} -y_ {i}) \ right | \ = \ \ left | \ sum _ {i = 1} ^ {n} (x_ {i} \ cdot y_ {i + 1} -y_ {i} \ cdot x_ {i + 1 }) \ right |}$.

Here, the indices that are greater than are always considered modulo , i.e. what is meant by: ${\ displaystyle n}$ ${\ displaystyle n}$${\ displaystyle x_ {n + 1}}$${\ displaystyle x_ {1}}$

${\ displaystyle 2A \ = \ \ left | x_ {n} \ cdot y_ {1} -y_ {n} \ cdot x_ {1} + \ sum _ {i = 1} ^ {n-1} (x_ {i } \ cdot y_ {i + 1} -y_ {i} \ cdot x_ {i + 1}) \ right |}$

In determinant form, the Gaussian trapezoidal formula is:

${\ displaystyle 2A \ = \ \ left | \ quad {\ begin {vmatrix} x_ {n} & y_ {n} \\ x_ {1} & y_ {1} \ end {vmatrix}} + \ sum _ {i = 1 } ^ {n-1} {\ begin {vmatrix} x_ {i} & y_ {i} \\ x_ {i + 1} & y_ {i + 1} \ end {vmatrix}} \ quad \ right |}$

In addition to the Gaussian trapezoidal formula, the area of ​​a polygon can be calculated using a signed sum of the areas of triangles that are formed with the edges of the polygon as the base and a fixed point (e.g. the point of origin) as the apex. The areas of the triangles with a base facing away from the fixed point (as the edge of the polygon) are given a negative sign.

The area of ​​lattice polygons whose corners are all on one lattice can be calculated with Pick's theorem .

## use

In computer science , important approximations of complex polygons are the convex hull and the minimally surrounding rectangle . In algorithms, a possible non-empty intersection with another geometric object is often first tested (or this is excluded) on the basis of the approximation, only then is the entire polygon loaded into memory and an exact intersection is calculated.

In 3D computer graphics , in addition to other methods of geometric modeling, any (including curved) surfaces are modeled as a polygon mesh . Triangle meshes are particularly suitable for the quick display of surfaces, but they cannot be interpolated as well using subdivision surfaces . There are a number of known data structures for storing polygonal networks.

Regular polygons are often used as a floor plan in architecture. Well-known examples: